Biological Networks (Complex Systems and Interdisciplinary Science)

BIOLOGICAL NETWORKS

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Complex Systems and Interdisciplinary Science (ISSN: 1793-4540)

Series Editors: Felix Reed-Tsochas (University of Oxford, UK) Neil Johnson (University of Oxford, UK) Associate Editors: Brian Arthur Santa Fe Institute, Spain

Philip Maini University of Oxford, UK

Robert Axtell George Mason University, USA

Martin Nowak Harvard University, USA

Stefan Bornholdt University of Bremen, Germany

Ricard Solé Santa Fe Institute, Spain

Janet Efstathiou University of Oxford, UK

Dietrich Stauffer University of Cologne, Germany

Pak Ming Hui The Chinese University of Hong Kong, China

Kagan Tumer Oregon State University, USA

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A Nasdaq Market Simulation: Insights on a Major Market from the Science of Complex Adaptive Systems by Vincent Darley & Alexander V. Outkin

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Large Scale Structure and Dynamics of Complex Networks edited by Alessandro Vespignani & Guido Caldarelli

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Biological Networks edited by François Képès

Forthcoming: Coping with Complexity: Understanding and Managing Complex Agent-Based Dynamical Networks edited by Janet Efstathiou, Neil F. Johnson & Felix Reed-Tsochas

Complex Systems and Interdisciplinary Science

Vol. 3

BIOLOGICAL NETWORKS

Editor

François Képès Genopole®, CNRS & University of Evry, France

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BIOLOGICAL NETWORKS Complex Systems and Interdisciplinary Science — Vol. 3 Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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PREFACE

Over the last few years, biologists have accumulated at unprecedented pace huge datasets on systems at many different levels, ranging from molecules to populations. As these datasets typically consisted of a list of biological objects and of their interactions, they could naturally be captured by network representations. These uniform representations allowed any single domain of application to benefit from scientific breakthroughs originating in several disciplines, from graph theory to technological or social networks. This Book on Biological Networks testifies to the recent efficiency of this transversal approach, while anticipating that the advent of more sophisticated types of abundant data may inspire combinations of network methods with other approaches in an application-driven fashion. Challenges In network models, the relevant components in a system are identified as nodes. The interactions between these components are represented as links between nodes. Following this abstraction step, it becomes possible to study the topological properties of the network thus obtained. The generality and uniformity of the network representation make it possible to compare systems of very different types. At present, pure and combined network-based approaches still present fascinating challenges with respect to topological properties, and to temporal and spatial development. Transversal challenges (Fig. 1) include tackling networks with a high number of nodes, and partitioning them and recomposing their parts in a useful way. Partitioning a network into sub-networks is of interest only if the resulting sub-networks or modules are biologically relevant and display a characteristic dynamics, that they retain upon recomposition into the full picture. While partitioning into relevant modules has met with some success, it is noteworthy that network recomposition is still in its infancy. Recomposition is however a requisite for the full success of

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the modular approach in fruitfully compacting representations and in building a knowledge of Nature's own design principles. Modularity has an evolutionary counterpart. In particular, it is quite possible that modularity contributes to the evolvability of organisms. In this respect, the relation between functional modules and evolutionary modules must be questioned. Another important transversal challenge to network-based approaches (Fig. 1) consists in tackling heterogeneity, a central feature of most empirical networks. This implies the possibility of expressing an arbitrary number of link types among a single set of nodes (layered networks), or an arbitrary number of node types, or heterogeneities both in nodes and links. Besides heterogeneities of components and links, it will also be fruitful to deal with temporal heterogeneity, i.e. connections that individually vary over time, even though they remain priviledged, specific interactions.

Figure 1. Challenges of biological networks.

In any realistic model, one would ideally like to unfold networks both in time (dynamics) and in space, to escape the static view that one often associates with network topological descriptions (Fig. 1). Among the shortcomings of current purely network-based approaches, most

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conspicuous is the lack of a geometrical space where to place the biological objects in neighborhood situations with potentially important effects on the dynamics of their interactions, from cellular regulations to epidemics. Indeed, many studies in complex systems follow either a network-based or an agent-based approach. Seldom are these approaches used jointly. The reasons for disjoint use typically include the requirement for simplicity in the modeling process. However, these two approaches may rather be seen as complementary. Indeed, space and locality are at the heart of agent-based approaches. However, these models fall short of allowing for perennial relations between specific agents, such as those encountered among individuals, cells and even biomolecules. Network models allow such priviledged interactions between specific agents. Thus, it would be useful to harness the networkbased and the agent-based approaches together for more realistic modeling that would involve both the movement of agents in space and priviledged interactions between specific agents. Finally, the search for design principles afforded by network approaches to biology will logically lead to applying these principles to build new forms of synthetic Life, for engineering purposes as well as for a better understanding of natural Life. Outline Part 1 of the Book addresses transversal topics, thus covering generic issues and providing the mathematical setting. The Chapters in Part 1 each survey many types of networks with a common question, although they anchor their discussion in a few well-chosen case studies for the sake of understanding. The common questions are topology in Chapter 1 by Barabási and coworkers, modularity in Chapter 2 by Solé and coworkers, and reverse engineering in Chapter 3 by d'Alché-Buc. Part 2 of the Book addresses vertical topics, thus covering in-depth various application domains at the molecular, cellular and population levels. Each such Chapter corresponds to one application domain, describing one type of network, from the interacting partners and discovery methods, to the overall structure, dynamics and modeling. At the molecular level, transcriptional networks are discussed in Chapter 4

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by Képès, protein networks in Chapter 5 by Ideker and coworker, metabolic networks in Chapter 6 by Fell, and mixed molecular networks in Chapter 7 by Schächter. At the cellular level, neuronal networks are covered in Chapter 9 by Frégnac and coworkers, and immunological networks in Chapter 10 by Callard and Stark. At the population level, Chapter 11 by Bersier offers a historical perspective and a wide introduction to ecological studies, while Chapter 12 by Martinez and coworker provides detailed innovative views on food webs. Finally, epidemiological networks are discussed in Chapter 13 by Koopman. A biological network cannot be fully understood unless the evolutionary dimension is considered. This is why counterparts on natural genesis or artificial generation of networks are provided when sufficient ground is available, notably for molecular networks in Chapter 8 by Bornberg-Bauer and coworkers, and to a lesser extent for ecological networks inside Chapter 11 by Bersier. Evolutionary considerations are not absent from the topical chapters though. Acknowledgements This Book is part of a nascent Series that World Scientific devotes to "Complex Systems and Inter-disciplinary Science". Without the exceptional dedication of Felix Reed-Tsochas from Oxford University, one of the Series Editor, this Book would have never seen the light of day. When Felix approached me with this idea, my response was immediately enthusiastic, as I had suffered in the past of the unavailability of a Book that would cover biological networks with both transversal spirit and in-depth insight, exactly what this Series was calling for. In preparing the Book project, his advices and encouragements have been particularly useful. It has been a pleasure to interact with Authors originating from a variety of disciplines and countries, some I knew well, some I am still hoping to meet in person some day, all of them enthusiastic scientists. I have learnt a lot by reading their contributions, and I wish that the Reader will learn from these chapters and enjoy them as much as I did. Over the last ten years many people have influenced and supported my exploration of biological networks. In particular, the Genopole®

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workgroups have been since their inception, and still are a constant and lively source of scientific inspiration for me and my close colleagues. I am very grateful to Lizzie Bennett from Imperial College Press in London, who provided support and advice at crucial moments. Last but not least, I gratefully acknowledge the editorial assistance of Sylvie Bobelet from the Epigenomics Project in Évry. Week after week, she bravely fought with reference mismatches and formatting issues which gave her a hard time, to finally assemble this Book.

Évry, March 2007

François Képès

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CONTENTS

Preface

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Contributors

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Chapter 1

Scale-Free Networks in Biology Eivind Almaas, Alexei Vázquez and Albert-László Barabási

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Chapter 2

Modularity in Biological Networks Ricard V. Solé, Sergi Valverde and Carlos Rodriguez-Caso

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Chapter 3

Inference of Biological Regulatory Networks: Machine Learning Approaches Florence d'Alché-Buc

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Chapter 4

Transcriptional Networks François Képès

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Chapter 5

Protein Interaction Networks Kai Tan and Trey Ideker

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Chapter 6

Metabolic Networks David A. Fell

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Chapter 7

Heterogeneous Molecular Networks Vincent Schächter

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Chapter 8

Evolution of Regulatory Networks Amélie Veron, Dion Whitehead and Erich Bornberg-Bauer

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Chapter 9

Contents

Complexity in Neuronal Networks Yves Frégnac, Michelle Rudolph, Andrew P. Davison and Alain Destexhe

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Chapter 10 Networks of the Immune System Robin E. Callard and Jaroslav Stark

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Chapter 11 A History of the Study of Ecological Networks Louis-Félix Bersier

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Chapter 12 Dynamic Network Models of Ecological Diversity, Complexity, and Nonlinear Persistence Richard J. Williams and Neo D. Martinez

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Chapter 13 Infection Transmission through Networks James S. Koopman

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Index

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CONTRIBUTORS

Eivind Almaas Center for Network Research Department of Physics University of Notre Dame, USA

Alain Destexhe Unité de Neurosciences Intégratives et Computationnelles (UNIC), Gif-sur-Yvette, France

Albert-László Barabási Center for Network Research Department of Physics, University of Notre Dame, USA

David A. Fell School of Life Sciences, Oxford Brookes University, UK Yves Frégnac Unité de Neurosciences Intégratives et Computationnelles (UNIC), Gif-sur-Yvette, France

Louis-Félix Bersier Unit of Ecology & Evolution, Fribourg University, Fribourg, Switzerland

Trey Ideker Department of Bioengineering University of California at San Diego, USA

Erich Bornberg-Bauer Division of Bioinformatics, Institute for Evolution and Biodiversity, The Westphalian Wilhelm's University of Münster, Germany

François Képès Epigenomics Project, Genopole®, CNRS, University of Évry, France

Robin E. Callard Immunobiology Unit, Institute of Child Health and CoMPLEX, University College London, UK

James S. Koopman Dept. of Epidemiology, University of Michigan, USA Neo D. Martinez Pacific Ecoinformatics and Computational Ecology Lab, USA

Florence d'Alché-Buc Informatique, Biologie Intégrative et Systèmes Complexes, CNRS & Epigenomics Project, Genopole®, Evry, France

Carlos Rodriguez-Caso Complex Systems Lab, ICREA, Universitat Pompeu Fabra, Spain Santa Fe Institute, New Mexico, USA

Andrew P. Davison Unité de Neurosciences Intégratives et Computationnelles (UNIC), Gif-sur-Yvette, France xiii

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Contributors

Michelle Rudolph Unité de Neurosciences Intégratives et Computationnelles (UNIC), Gif-sur-Yvette, France

Alexei Vázquez Center for Network Research Department of Physics, University of Notre Dame, USA

Vincent Schächter Genoscope, CEA, CNRS, Evry, France

Amélie Veron Division of Bioinformatics, Institute for Evolution and Biodiversity, The Westphalian Wilhelm's University of Münster, Germany

Ricard V. Solé Complex Systems Lab, ICREA, Universitat Pompeu Fabra, Spain, and Santa Fe Institute, New Mexico, USA Jaroslav Stark CISBIC and Department of Mathematics, Imperial College London, UK Kai Tan Department of Bioengineering University of California at San Diego, USA Sergi Valverde Complex Systems Lab, ICREA, Universitat Pompeu Fabra, Spain, and Santa Fe Institute, New Mexico, USA

Dion Whitehead Division of Bioinformatics, Institute for Evolution and Biodiversity, The Westphalian Wilhelm's University of Münster, Germany Richard J. Williams Microsoft Research Ltd, Cambridge, UK

CHAPTER 1 SCALE-FREE NETWORKS IN BIOLOGY

Eivind Almaas, Alexei Vázquez and Albert-László Barabási Center for Network Research and Department of Physics, University of Notre Dame, Notre Dame, IN 46556,USA [email protected], [email protected], [email protected]

1. Introduction The last century brought with it unprecedented technological and scientific progress, rooted in the success of the reductionist approach. For many current scientific problems, however, it is not possible to predict the behavior of a system from an understanding of its (often identical) elementary constituents and their individual interactions. For these systems we need to develop new methods in order to gain insight into their properties and dynamics. During the last few years network approaches have shown great promise in this direction, offering new tools to analyze and understand a host of complex systems (1-7). A much studied example concerns communication systems like the internet and the world wide web, which are modeled as networks with nodes being the routers (8) or web pages (9) and the links are the physical wires or URL’s, respectively. The network approach also lends itself to the analysis of societies, with people as nodes and the connections between the nodes representing friendships (10), collaborations (11,12), sexual contacts (13) or co-authorship of scientific papers (14,15) to name a few possibilities. It seems that the more we scrutinize the world surrounding us, the more we realize that we are hopelessly entangled in myriads of

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interacting webs, and to describe them we need to understand the architecture of the various networks that nature and technology offers us. Biological systems ranging from food webs in ecology to biochemical interactions in molecular biology can benefit greatly from being analyzed as networks. In particular within the cell the variety of interactions between genes, proteins and metabolites are well captured by network representations, especially with the availability of veritable mountains of interaction data from genomics approaches. In this Chapter we will discuss recent results and developments in the study and characterization of the structure and utilization of biological networks. 2. Characterizing Network Topology There are by now many tools and measures available to study the structure of complex networks. In the following we will discuss three of the most fundamental quantities, the degree distribution, node clustering and hierarchy, and the issue of subgraphs and motifs. In addition, it is customary to investigate the betweenness-centrality (BC) of both nodes and links, and the network assortativity. The BC is related to the number of shortest paths going through either a node or a link, and hence a large BC value indicates that the node or link acts as a bridge by connecting different parts of the network (16). The assortativity describes the propensity of a node to be directly connected to other nodes with similar degree (17,18). 2.1. Degree Distribution The representation of various complex systems as networks has revealed surprising similarities, many of which are intimately tied to power laws. The simplest network measure is the average number of nearest neighbors of a node, or the average degree k . However, this is a rather crude property, and to gain further insight into the topological organization of real networks, we need to determine the variation in the nearest neighbors, given by the degree distribution P (k ) . For a surprisingly large number of networks, this degree distribution is best characterized by the power law functional form (19) (Fig.1a);

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P(k) ~ k −γ .

(1)

Important examples include the metabolic network of 43 organisms (20), the protein interaction network of S. cerevisiae (21) C. elegans (22), D. melanogaster (23), and various food webs (24). If the degree distribution instead was single-peaked (e.g. Poisson or Gaussian) as in Fig. 1b, the majority of the nodes would be well described by the average degree and we can with reason talk about a “typical” node of the network. This is very different for networks with a power-law degree distribution; the majority of the nodes only have a few neighbors, while many nodes have hundreds and some even thousands of neighbors. Although average node degree values can be calculated for these networks since their size is finite, these values are not representative of a typical node. For this reason, these networks are often referred to as “scale-free”. (a)

(b)

Figure 1. Characterizing degree distributions. For the power-law degree distribution (a), there exists no typical node, while for single peaked distributions (b), most nodes are well represented by the average (typical) node with degree k .

2.2. Clustering Coefficient A measure that gives insight into the local structure of a network is the so-called clustering of a node: the degree to which the neighborhood of a node resembles a complete subgraph (25).

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For a node i with degree k i the clustering is defined as

Ci =

2 ni , k i (k i − 1)

(2)

representing the ratio of the number of actual connections between the neighbors of node i to the number of possible connections. For a node which is part of a fully interlinked cluster C i = 1 , while C i = 0 for a node where none of its neighbors are interconnected. Accordingly, the overall clustering coefficient of a network with N nodes is given by C = C i / N , quantifying a network’s potential modularity. By

∑

studying the average clustering of nodes with a given degree k, information about the actual modular organization of a network can be extracted (26-29): For all metabolic networks available, the average clustering follows a power-law form as

C(k) ~ k −α ,

(3)

suggesting the existence of a hierarchy of nodes with different degrees of modularity (as measured by the clustering coefficient) overlapping in an iterative manner (26). In summary, we have seen strong evidence that biological networks are both scale-free (20,21) and hierarchical (26). 2.3. Subgraphs and Motifs A number of complex biological and non-biological networks were recently found to contain network motifs, representing elementary interaction patterns between small groups of nodes (subgraphs) that occur substantially more often than would be expected in a random network of similar size and connectivity (1,2). Theoretical and experimental evidence indicates that at least some of these recurring elementary interaction patterns carry significant information about the given network’s function and overall organization (30-33). For example, transcriptional regulatory networks of cells (30,31,34,35 ; see Chapter 4), neural networks of C. elegans and some electronic circuits (31) are all information processing networks that contain a significant number of feed-forward loop motifs (see Chapter 2). However, in transcription-

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regulatory networks these motifs do not exist in isolation but meld into motif clusters (36), while other networks are devoid of feed-forward loops altogether (31).

Figure 2. The phase diagrams organize the subgraphs based on the number of nodes (n, horizontal axis) and the number of links (m, vertical axis), each discrete point explicitly depicting the corresponding subgraph. The stepped yellow line corresponds to the predicted phase boundary separating the abundant Type I subgraphs (below the line) from the constant density Type II subgraphs (above the line). The background color is proportional to the relative subgraph count Cnm=Nnm/ΣsNns of each n-node subgraph, the color code being shown in the upper right corner. Note that some (n,m) points in the phase diagram may correspond to several topologically distinguishable subgraphs. For simplicity, we depict only one representative topology in such cases. As the yellow phase boundary depends on the γ and α exponents of the corresponding network, each phase diagram is slightly different. Yet, there is a visible similarity between the networks of the same kind: the phase diagrams of the two transcription or the two metabolic networks are almost indistinguishable.

The number Nnm of subgraphs with n nodes and m interactions expected for a network of N nodes can be estimated from the two key topological parameters of a network’s large-scale structure: the degree exponent, γ, and the hierarchical exponent. In general we find that there are two subgraph classes: Type I subgraphs are those that satisfy (m-n+1)α-(n-γ)0, and their number is given by NIInm~N. As even for finite networks kmax>>1, the typical number of Type I subgraphs is significantly larger than the number of Type II subgraphs (NInm/NIInm >>1). Moreover, for infinite systems (N→∞) the relative number of Type II subgraphs is vanishingly small compared to Type I subgraphs, as NInm/NIInm→∞. This subdivision in Type I and II subgraphs is shown in Fig. 2 for five cellular networks: the metabolic networks of E. coli and S. cerevisiae, the regulatory networks of E. coli and S. cerevisiae, and the protein interaction network of S. cerevisiae; and different (n,m) subgraphs. The (m-n+1)α-(n-γ)=0 condition, predicted to separate the Type I and II subgraphs, appears as stepped yellow phase boundaries in the phase diagrams. For example, for the E. coli transcriptional regulatory network with α=1 and γ=2.1 (Table 1) the phase boundary corresponds to a stepped-line with approximate overall slope 1+1/α=2.0 and intercept -1γ/α =-3.1 (Fig. 1a). The Type II subgraphs are those above this boundary, and should be either absent, or present only in very low numbers in the transcriptional regulatory network. In contrast, the Type I subgraphs below the boundary are predicted to be abundant. Comparing Figs. 2a-e we find that while the stepped phase boundaries for the different cellular networks differ due to the differences in the (γ,α) exponents (Table 1), the observed densities in the real networks follow relatively closely the predicted phase boundaries. Occasional local deviations from the predictions can be attributed to the error bars of the (γ,α) exponents (Table 1), which allow for some local uncertainties for the phase boundary. Figures 1a-e also indicate that, in agreement with the empirical findings (30-33), each cellular network is characterized by a distinct set of over-represented Type I subgraphs, raising the possibility of classifying networks based on their local structure (4). Yet, the phase diagrams demonstrate that knowledge of the two global topological parameters introduced in Sections 2.1 and 2.2 automatically uncovers the local structure of cellular networks, suggesting that a subgraph- or motifbased classification are equivalent with a classification based on the different (γ,α) exponents characterizing these networks.

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Figure 3. Graphical representation of three network models: (a) and (d) The ER (random) model, (b) and (e) the BA (scale-free) model and (c) and (f) the hierarchical model. The random network model is constructed by starting from N nodes before the possible nodepairs are connected with probability p. Panel (a) shows a particular realization of the ER model with 10 nodes and connection probability p=0.2. In Panel (b) we show the scalefree model at time t (green links) and at time (t+1) when we have added a new node (red links) using the preferential attachment probability (see Eq. (4)). Panel (c) demonstrates the iterative construction of a hierarchical network, starting from a fully connected cluster of four nodes (blue). This cluster is then copied three times (green) while connecting the peripheral nodes of the replicas to the central node of the starting cluster. By once more repeating this replication and connection process (red nodes), we end up with a 64-node scale-free hierarchical network. In Panel (d) we display a larger version of the random network, and it is evident that most nodes have approximately the same number of links. For the scale-free model, (e) the network is clearly inhomogeneous: while the majority of nodes has one or two links, a few nodes have a large number of links. We emphasize this by coloring the five nodes with the highest number of links red and their first neighbors green. While in the random network only 27% of the nodes are reached by the five most connected nodes, we reach more than 60% of the nodes in the scale-free network, demonstrating the key role played by the hubs. Note that the networks in (d) and (e) consist of the same number of nodes and links. Panel (f) demonstrates that the standard clustering algorithms are not that successful in uncovering the modular structure of a scale-free hierarchical network.

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Table 1. The γ and α exponents for five cellular networks, determined from a direct fit to the P(k) and C(k) functions.

γ α

Transcription E. coli S. cerevisiae 2.1±0.3 2.0±0.2 1.0±0.2 1.0±0.2

Metabolic E. coli S. cerevisiae 2.0±0.4 2.0±0.1 0.8±0.3 0.7±0.3

Protein Interaction S. cerevisiae 2.4±0.4 1.3±0.5

3. Network Models As we have just seen, many biological networks are dominated by a scale-free distribution of nearest neighbors. Why is this power-law behavior so pervasive? To understand the cause of the scale-free degree distribution and the hierarchical network structure, we will in the following explain three models that serve as network paradigms. These models build on very different principles and, to varying degrees, are able to explain the observed network features.

Figure 4. Properties of the three network models. (a) The ER model gives rise to a Poisson degree distribution P(k) (the probability that a randomly selected node has exactly k links) which is strongly peaked at the average degree k . The degree distributions for the scale-free (b) and the hierarchical (c) network models do not have a peak, they instead decay according to P(k ) ~ k −γ . The average clustering coefficient for nodes with exactly k neighbors, C(k), is independent of k for both the ER (d) and the scale-free (e) network model. (f) In contrast, C(k)~k −1 for the hierarchical network model.

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3.1. Random Network Model In discussing the origin of the observed power-law behavior, we need to first understand the properties of the simplest available network model. While graph theory initially focused on regular graphs, since the 1950's large networks with no apparent design principles were described as random graphs (37), proposed as the simplest and most straightforward realization of a complex network. According to this Erdos−Renyi (ER) model of random networks (38), we start with N nodes and connect every pair of nodes with probability p. This creates a graph with approximately pN(N-1)/2 randomly distributed edges (Fig. 3a,d). The distribution of nearest neighbors follows a Poisson distribution (Fig. 4a), and consequently the average degree k of the network describes the properties of a typical node. Furthermore, for this “democratic” network model, the clustering is independent of the node degree k (Fig. 4d). The ER model, although simple and appealing, does not capture the properties of neither the degree distribution nor the clustering coefficient observed in biological networks. 3.2. Scale-Free Network Model In the network model of Barabási and Albert (BA), two key mechanisms, which both are absent from the classical random network model, are responsible for the emergence of a power-law degree distribution (19). First, networks grow through the addition of new nodes linking to nodes already present in the system. Second, there is a higher probability to link to a node with a large number of connections, a property called preferential attachment. These two principles are implemented as follows: starting from a small core graph consisting of m0 nodes, a new node with m links is added at each time step and connected to the already existing nodes (Fig. 3b,e). Each of the m new links are then preferentially attached to a node i (with ki neighbors) which is chosen according to the probability

Π i = ki / ∑ k j . j

(4)

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The simultaneous combination of these two network growth rules gives rise to the observed power-law degree distribution (Fig. 4b). In Panel 3b, we illustrate the growth process of the scale-free model by displaying a network at time t (green links) and then at time (t+1), when we have added a new node (red links) using the preferential attachment probability. Compared to random networks, the probability that a node is highly connected is statistically significant in scale-free networks. Consequently, many network properties are determined by a relatively small number of highly connected nodes, often called “hubs”. To make the effect of the hubs on the network structure visible, we have colored the five nodes with largest degrees red in Fig. 3d and 3e and their nearest neighbors green. While in the ER network only 27% of the nodes are reached by the five most connected ones, we reach more than 60% of the nodes in the scale-free network, demonstrating the key role played by the hubs. Another consequence of the hub’s dominance of the network topology is that scale-free networks are highly tolerant of random failures (perturbations) while being extremely sensitive to targeted attacks (39). Comparing the properties of the BA network model with those of the ER model, we note that the clustering of the BA network is larger, however C (k ) is approximately constant (Fig. 4e), indicating the absence of a hierarchical structure. 3.3. Hierarchical Network Model Many real networks are expected to be fundamentally modular, meaning that the network can be seamlessly partitioned into a collection of modules where each module performs an identifiable task, separable from the function(s) of other modules (40-43 ; see Chapter 2). Therefore, we must reconcile the scale-free property with potential modularity. In order to account for the modularity as reflected in the power-law behavior of C (k ) and a simultaneous scale-free degree distribution, we have to assume that clusters combine in an iterative manner, generating a hierarchical network (26,29). Such a network emerges from a repeated duplication and integration process of clustered nodes (26), which in principle can be repeated indefinitely. This process is depicted in Panel 2c, where we start from a small cluster of four densely linked

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nodes (blue). We next generate three replicas of this hypothetical initial module (green) and connect the three external nodes of the replicated clusters to the central node of the old cluster, thus obtaining a large 16-node module. Subsequently, we again generate three replicas of this 16-node module (red), and connect the 16 peripheral nodes to the central node of the old module, obtaining a new module of 64 nodes. This hierarchical network model seamlessly integrates a scale-free topology with an inherent modular structure by generating a network that has a power law degree distribution (Fig. 4c) with degree exponent γ = 1 + ln 4 / ln 3 ≈ 2.26 and a clustering coefficient C(k) which proves to be dependent on k −1 (Fig. 4f). However, note that modularity does not imply clear-cut sub-networks linked in well-defined ways (26,44). In fact, the boundaries of modules are often blurred considerably (see e.g. Fig. 3f). 3.4. Bose-Einstein Condensation and Networks In most complex systems the nodes have differing abilities of attracting new links, which is independent of their number of nearest neighbors. For instance, some Web pages quickly acquire a large number of links through a mixture of good content and marketing, although they are just recently published on the World wide web. This competition for links can be incorporated into the scale-free model by adding a "fitness" parameter, ηi, to each node, i, describing its ability to compete for links at the expense of other nodes. For example, a Web page with good up-todate content and a friendly interface would be expected to display a greater fitness than a low-quality page that is only updated occasionally. The probability Πi that a new node connects to one with ki links is then modified from Eq. (4) such that Πi = ηi ki/Σj ηj kj (45). The competition generated by the various fitness levels means that each node evolves differently in time compared with others. Indeed, the connectivity of each node is now given by ki(t) ~ tß(η), where the exponent ß(η) increases with η, and t is the time since the node was added to the network (45). Consequently, fit nodes (ones with large η) can join the network at some later time and connect to many more links than less-fit nodes that have been around for longer.

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Amazingly, such competitive-fitness models appear to have close ties with Bose-Einstein condensation, currently one of the most investigated problems in atomic physics. In a normal atomic gas, the atoms are distributed among many different energy levels. However in a BoseEinstein condensate, all the atoms accumulate in the lowest energy state of the system and are described by the same quantum wave function. By replacing each node in the network with an energy level having energy εi= exp(-β ηi), the fitness model maps exactly onto a Bose gas (45). According to this mapping, the nodes map to energy levels while the links are represented by atoms in these levels. Additionally, the behavior of a Bose gas is uniquely determined by the distribution g(ε) from which the random energy levels (or fitnesses) are selected. One expects that the functional form of g(ε) depends on the system. For example, the attractiveness of a router to a network engineer comes from a rather different distribution than the fitness of a dot-com company competing for customers. For a wide class of g(ε) distributions, a "fit-get-richer" phenomenon emerges (45). Although the fittest node acquires more links than its lessfit counterparts, there is no clear winner. On the other hand, certain g(ε) distributions can result in a Bose-Einstein condensation, where the fittest node does emerge as a clear winner. For these distributions, a condensate develops by acquiring a significant fraction of the links which is independent of the size of the system. In network language this corresponds to a "winner-takes-all" phenomenon. While the precise form of the fitness distribution for the Web or the Internet is not known yet, it is likely that g(ε) could be measured in the near future. 4. Network Utilization Despite their impressive successes, purely topologic approaches have important intrinsic limitations. For example, the activity of the various metabolic reactions or regulatory interactions differs widely, some being highly active under most growth conditions while others are switched on only for some rare environmental circumstances. Therefore, an ultimate description of cellular networks requires us to consider the intensity (i.e., strength), the direction (when applicable) and the temporal aspects of the

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interactions. While we so far know little about the temporal aspects of the various cellular interactions, recent results have shed light on how the strength of the interactions is organized in metabolic and geneticregulatory networks (46-48) and how the local network structure is correlated with these link strengths. 4.1. Flux Utilization In metabolic networks the flux of a given metabolic reaction, representing the amount of substrate being converted to a product within unit time, offers the best measure of interaction strength. Recent advances in metabolic flux-balance approaches (FBA, see also Chapter 6) (49-52) allow us to calculate the flux for each reaction, and they have significantly improved our ability to generate quantitative predictions on the relative importance of the various reactions, thus leading to experimentally testable hypotheses. The FBA approaches can be described as follows: Starting from a stoichiometric matrix model of an organism, e.g. one for E. coli contains 537 metabolites and 739 reactions (49-51), the steady state concentrations of all metabolites must satisfy

d [ Ai ] = ∑ Sij ν j = 0 dt j

(5)

where S ij is the stoichiometric coefficient of metabolite Ai in reaction j and ν j is the flux of reaction j. We use the convention that if metabolite Ai is a substrate (product) in reaction j, S ij < 0 ( S ij > 0 ), and we constrain all fluxes to be positive by dividing each reversible reaction into two “forward” reactions with positive fluxes. Any vector of positive fluxes {ν j } which satisfies Eq. (5) corresponds to a state of the metabolic network, and hence, a potential state of operation of the cell. Assuming that the cellular metabolism is in a steady state and optimized for the maximal growth rate (50,51), FBA allows us to calculate the flux for each reaction using linear optimization, providing a measure of each reaction’s relative activity (46). A striking feature of the resulting flux distribution from such modeling of both H. pylori, E. coli

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and S. cerevisiae is its overall inhomogeneity: reactions with fluxes spanning several orders of magnitude coexist under the same conditions (Fig. 5a). This is captured by the flux distribution for E. coli, which follows a power law where the probability that a reaction has flux ν is given by P (ν ) ~ (ν + ν 0 ) −α . This flux exponent is predicted to be α = 1.5 by FBA methods (46). In a recent experiment (53) the strength of the various fluxes of the E. coli central metabolism was measured, revealing (46) the power-law flux dependence P (ν ) ~ ν −α with α ≅ 1 (Fig. 5b). This power law behavior indicates that the vast majority of reactions have quite small fluxes, while coexisting with a few reactions with extremely large flux values.

(a)

(b)

Figure 5. Flux distribution for the metabolism of E. coli. (a) Flux distribution for optimized biomass production on succinate (black) and glutamate (red) rich uptake substrates. The solid line corresponds to the power law fit P (ν ) ~ (ν + ν 0 ) −α with ν 0 = 0.0003 and α = 1.5 . (b) The distribution of experimentally determined fluxes (53) from the central metabolism of E. coli also displays power-law behavior with a best fit to P(ν ) ~ ν −α with α = 1 .

The observed flux distribution is compatible with two quite different potential local flux structures (46). A homogeneous local organization would imply that all reactions producing (consuming) a given metabolite have comparable fluxes. On the other hand, a more delocalized “hot backbone” is expected if the local flux organization is heterogeneous, such that each metabolite has a dominant source (consuming) reaction. To distinguish between these two scenarios for each metabolite i produced (consumed) by k reactions, we define the measure (54,55)

Scale-Free Networks in Biology

⎛ νˆ ij Y (k , i ) = ∑ ⎜ k ⎜ j =1 ∑ νˆil ⎝ l =1 k

⎞ ⎟ ⎟ ⎠

15

2

,

(6)

where νˆij is the mass carried by reaction j which produces (consumes) metabolite i. If all reactions producing (consuming) metabolite i have comparable νˆij values, Y (k , i ) scales as 1 / k . If, however, a single reaction’s activity dominates Eq. (6), we expect Y (k , i ) ~ 1 , i.e., Y (k , i ) is independent of k. For the E. coli metabolism optimized for succinate and glucose uptake we find that both the in and out degrees follow the power law Y (k , i ) ~ k −0.27 , representing an intermediate behavior between the two extreme cases (46). This suggests that the large-scale inhomogeneity observed in the overall flux distribution is increasingly valid at the level of the individual metabolites as well: for most metabolites, a single reaction carries the majority of the flux. Hence, the majority of the metabolic flux is carried along linear pathways – the metabolic high flux backbone (HFB) (46). 4.2. Gene Interactions One can also investigate the strength of the various genetic regulatory interactions provided by microarray datasets. Assigning each pair of genes a correlation coefficient which captures the degree to which they are co-expressed, one finds that the distribution of these pair-wise correlation coefficients follows a power law (47,48). That is, while the majority of gene pairs have only weak correlations, a few gene pairs display a significant correlation coefficient. These highly correlated pairs likely correspond to direct regulatory and protein interactions. This hypothesis is supported by the finding that the correlations are larger along the links of the protein interaction network and between proteins occurring in the same complex than for pairs of proteins that are not known to interact directly (56-59). Taken together, these results indicate that the biochemical activity in both the metabolic and genetic networks is dominated by several ‘hot links’ that represent a few high activity interactions embedded into a web

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of less active interactions. This attribute does not seem to be a unique feature of biological systems: hot links appear in a wide range of nonbiological networks where the activity of the links follows a wide distribution (60,61). The origin of this seemingly universal property is, again, likely rooted in the network topology. Indeed, it seems that the metabolic fluxes and the weights of the links in some non-biological system (60,61) are uniquely determined by the scale-free nature of the network. A more general principle that could explain the correlation distribution data as well is currently lacking. 5. Conclusion Power laws are abundant in nature, affecting both the construction and the utilization of real networks. The power-law degree distribution has become the trademark of scale-free networks and can be explained by invoking the principles of network growth and preferential attachment. However, many biological networks are inherently modular, a fact which at first seems to be at odds with the properties of scale-free networks. However, these two concepts can co-exist in hierarchical scale-free networks. In the utilization of complex networks, most links represent disparate connection strengths or transportation thresholds. For the metabolic network of E. coli we can implement a flux-balance approach and calculate the distribution of link weights (fluxes), which (reflecting the scale-free network topology) displays a robust power-law, independent of exocellular perturbations. Furthermore, this global inhomogeneity in the link strengths is also present at the local level, resulting in a connected “hot-spot” backbone of the metabolism. Similar features are also observed in the strength of various genetic regulatory interactions. Despite the significant advances witnessed the last few years, network biology is still in its infancy, with future advances most notably expected from the development of theoretical tools, development of new interactive databases and increased insights into the interplay between biological function and topology.

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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

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23. Giot, L., Bader, J.S., Brouwer, C., Chaudhuri, A., Kuang, B., et al. A protein interaction map of Drosophila melanogaster. Science. 302, 1727. 24. Montoya, J.M. and Sole, R.V. (2002). Small-world patterns in food webs. J Theor Biol. 214, 405-12. 25. Watts, D.J. and Strogatz, S.H. (1998). Collective dynamics of small-world networks. Nature. 393, 440-2. 26. Ravasz, E., Somera, A.L., Mongru, D.A., Oltvai, Z.N. and Barabási, A.L. (2002). Hierarchical organization of modularity in metabolic networks. Science. 297, 15515. 27. Ravasz, E. and Barabási, A.L. (2003). Hierarchical organization in complex networks. Phys Rev E. 67, 026112. 28. Dorogovtsev, S.N., Goltsev, A.V. and Mendes, J.F.F. (2002). Pseudofractal scalefree web. Phys Rev E. 65, 066122. 29. Vázquez, A., Pastor-Satorras, R. and Vespignani, A. (2002). Large-scale topological and dynamical properties of the Internet. Phys Rev E. 65, 066130. 30. Shen-Orr, S., Milo, R., Mangan, S. and Alon, U. (2002) Nat Genet. 31, 64-8. 31. Milo, R., Shen-Orr, S.S., Itzkovitz, S., Kashtan, N. and Alon, U. (2002). Science. 298, 824-27. 32. Mangan, S., Zaslaver, A. and Alon, U. (2003). J Mol Biol. 334, 197-204. 33. Milo, R., Itzkovitz, S., Kashtan, N., Levitt, R., Shen-Orr, S., Ayzenshtat, I., Sheffer, M. and Alon, U. (2004). Science. 303, 1538-42. 34. Lee, T.I., Rinaldi, N.J., Robert, F., Odom, D.T., Bar-Joseph, Z., et al. (2002). Science. 298, 799-804. 35. Hinman, V.F., Nguyen, A.T., Cameron, R.A. and Davidson, E.H. (2003). Proc Natl Acad Sci. U.S.A. 100, 13356-61. 36. Dobrin, R., Beg, Q.K., Barabási, A.L. and Oltvai, Z.N. (2004). BMC Bioinformatics. 5, 10. 37. Bollobas, B. (1985). Random Graphs. Academic Press, London. 38. Erdos, P. and Renyi, A. (1960). On the evolution of random graphs. Publ Math Inst Hung Acad Sci. 5, 17-61. 39. Albert, R., Jeong, H. and Barabási, A.L. (2000). Attack and error tolerance of complex networks. Nature. 406, 378-82. 40. Hartwell, L.H., Hopfield, J.J., Leibler, S. and Murray, A.W. (1999). From molecular to modular cell biology. Nature. 402, C47-52. 41. Rao, C.V. and Arkin, A.P. (2001). Control motifs for intracellular regulatory networks. Annu Rev Biomed Eng. 3, 391. 42. Hasty, J., McMillen, D., Isaacs, F. and Collins, J.J. (2001). Computational studies of gene regulatory networks: In numero molecular biology. Nature Rev Genet. 2, 26879. 43. Shen-Orr, S.S., Milo, R., Mangan, S. and Alon, U. (2001). Network motifs in the transcriptional regulation network of Escherichia coli. Nature Genet. 31, 64-8. 44. Holme, P., Huss, M. and Jeong, H. (2003). Subnetwork hierarchies of biochemical pathways. Bioinformatics. 19, 532-9.

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45. Bianconi, G. and Barabási. A.L. (2001). Bose-Einstein condensation in complex networks. Phys Rev Lett. 86, 5632. 46. Almaas, E., Kovacs, B., Vicsek, T., Oltvai, Z.N. and Barabási, A.L. (2004). Global organization of metabolic fluxes in the bacterium Escherichia coli. Nature. 427, 839. 47. Kutznetsov, V.A., Knott, G.D. and Bonner, R.F. (2002). General statistics of stochastic processes of gene expression in eukaryotic cells. Genetics. 161, 1321-32. 48. Farkas, I.J., Jeong, H., Vicsek, T., Barabási, A.L. and Oltvai, Z.N. (2003). The topology of the transcription regulatory network in the yeast, Saccharomyces cerevisiae. Physica A. 318, 601-12. 49. Edwards, J.S. and Palsson, B.O. (2000). The Escherichia coli MG1655 in silico metabolic genotype: its definition, characteristics, and capabilities. Proc Natl Acad Sci. 97, 5528-33. 50. Edwards, J.S., Ibarra, R.U. and Palsson, B.O. (2001). In silico predictions of Escherichia coli metabolic capabilities are consistent with experimental data. Nat Biotechnol. 19, 125-30. 51. Ibarra, R.U., Edwards, J.S. and Palsson, B.O. (2002). Escherichia coli K-12 undergoes adaptive evolution to achieve in silico predicted optimal growth. Nature. 420, 186-9. 52. Segre, D., Vitkup, D. and Church, G.M. (2002). Analysis of optimality in natural and perturbed metabolic networks. Proc Natl Acad Sci. 99, 15112-7. 53. Emmerling, M., Dauner, M., Ponti, A., Fiaux, J., Hochuli, M., Szyperski, T., Wuthrich, K., Bailey, J.E. and Sauer, U. (2002). Metabolic flux responses to pyruvate kinase knockout in Escherichia coli. J Bacteriol. 184, 152-64. 54. Barthelemy, M., Gondran, B. and Guichard, E. (2003). Spatial structure of the Internet traffic. Physica A. 319, 633-42. 55. Derrida, B. and Flyvbjerg, H. (1987). Statistical properties of randomly broken objects and of multivalley structures in disordered-systems. J. Phys. A: Math Gen. 20, 5273-88. 56. Dezso, Z., Oltvai, Z.N. and Barabási, A.L. (2003). Bioinformatics analysis of experimentally determined protein complexes in the yeast, Saccharomyces cerevisiae. Genome Res. 13, 2450-4. 57. Grogoriev, A. (2001). A relationship between gene expression and protein interactions on the proteome scale: analysis of the bacteriophage T7 and yeast Saccharomyces cerevisiae. Nucleic Acids Res. 29, 3513-9. 58. Jansen, R., Greenbaum, D. and Gerstein, M. (2002). Relating whole-genome expression data with protein-protein interactions. Genome Res. 12, 37-46. 59. Ge, H., Liu, Z., Church, G.M. and Vidal, M. (2001). Correlation between transcriptome and interactome mapping data from Saccharomyces cerevisiae. Nature Genet. 29, 482-6. 60. Goh, K.-I., Kahng, B. and Kim, D. (2002). Fluctuation-driven dynamics of the internet topology. Phys Rev Lett. 88, 108701. 61. de Menezes, M.A. and Barabási, A.L. (2004). Fluctuations in network dynamics. Phys Rev Lett. 92, 028701.

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CHAPTER 2 MODULARITY IN BIOLOGICAL NETWORKS

Ricard V. Solé, Sergi Valverde and Carlos Rodriguez-Caso Complex Systems Lab, ICREA-Universitat Pompeu Fabra, 08003 Barcelona, Spain, and Santa Fe Institute, 1399 Hyde Park Road, 08075 Santa Fe, New Mexico, USA [email protected], [email protected], [email protected]

1. Introduction The intimate structure of cellular life is largely associated to the networks of interactions among different types of molecules. The structure of such cellular and molecular networks, from the genome and the proteome to the protein folding graphs is known to be very heterogeneous and often reveals a characteristic modular architecture (1,2). At one level, it has been shown that most units (amino acids, genes, proteins or metabolites) are linked to a few other units but invariably a few units exhibit a large number of links. Such heterogeneity has been also found in a wide spectrum of other complex systems, from natural to artificial (3-6). More importantly, the topological organization of complex nets might pervade their efficiency, robustness and fragility under perturbations (7). Understanding the origins and meaning of these topological maps is an important step towards understanding the role played by different mechanisms of evolution (7). Modules have been found in biological systems at multiple levels, from RNA structure (8) to the cerebral cortex (see ref. 9) and references therein). The widespread character of modular organization pervades the

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functional association between compartmentalization and evolution. Modules have been variously defined as: (a) functionally buffered, (b) robust, (c) independently controlled, (d) plastic in composition and interconnectivity and (d) evolutionarily conserved. The evolutionary conservation of modules is clearly appreciated in gene networks involved in early development (10,11). The argument is that the special features of some of these modules are tightly linked to their robustness under different sources of noise. As discussed in Chapter 1, real biological networks are typically scale-free: their degree distributions fall off as a power law P(k) ~ k −γ . Real networks are also known to display the so-called small world effect, defined in terms of high clustering coefficient (high number of triangles compared to a randomly wired web with the same distribution of links) and very short path lengths. The presence of a high heterogeneity is actually tied to the small number of degrees of separation: the highly connected hubs seem to act as glue in these webs, allowing most nodes to connect to each other through a small number of jumps. The modular structure exhibited by biological webs is naturally associated to hierarchical properties (12). An example of modular network is shown in Fig. 1a : here a very simple picture is shown, involving three coupled modules. Each module has the same number of units and they are more connected within the module than with units outside it. The system shown here is generated as follows (12). Nodes inside each module are randomly wired with some probability p, as in so-called Erdös-Renyi (ER) graphs (13). They are also linked to nodes in other modules with a probability q1 not allowed) gave similar results to fixed parameter simulations (results not shown). Second, even though functional responses could be different for each link in the network (21), we specify a single value of qij or cij, so each link within a network is of the same type.

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Unless stated otherwise, we assume that predator species have equal preference (αij) for all their prey. If ni is the number of prey that species i consumes, αij = 1/ni for each species j in the diet of species i. We also systematically vary the αij of omnivores to examine the effects of skewing diets to higher or lower trophic level prey. The range of αij is defined by a preference skewness k = αimax/αimin, where αimax and αimin are the preferences for the prey items of species i with the maximum and minimum trophic level (TLmax and TLmin) respectively. For each prey species j of species i, we define bij = 1 + (k − 1)(TL j − TLmin ) /

(TLmax − TLmin ) , where TLj is the trophic level of prey item j.

The

preference of species i for prey item j is then α ij = bij / ∑ bil , where the l

sum is across all prey items of species i. When k = 1, all prey preferences of an omnivore are equal; when k < 1, low trophic level prey are preferred and when k > 1, high trophic level prey are preferred. Each simulation begins by building an initial random, cascade, or niche model web of a certain size (S0) and connectance (C0). The integrated structure-dynamic model then computes which species persist with positive biomass greater than an extinction threshold of 10-15 after 4000 time steps. Following any extinctions, a “persistent web” with SP species and connectance CP remains. As the structural models are stochastic, this procedure is repeated a large number of times so that statistical properties of the integrated structure-dynamic model is ascertained. Both the functional response control parameters and a predator’s preferences among prey are varied to study effects of foodweb dynamics on persistence and food-web structure. For each model iteration, we define absolute persistence PA = SP and relative persistence as PR = SP/S0. Overall persistence P is the mean value of PR across a set of iterations. Topological properties of the persistent webs were compared to different versions of niche webs. Here, we focus on the distribution of trophic levels and connectivity among species by examining the fractions of top, intermediate, basal, omnivorous, and herbivorous species, mean trophic level, and the standard deviation of the connectivity of each species.

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3. Topology and Dynamics

We analyzed the behavior of our dynamic network models with respect to the combined variation of several key parameters. The models’ high dimensionality prevents full examination of all the combinations of parameter values that were analyzed. Instead, we present a sequence of results that describes the effects of varying a few parameters and then fix these parameters and analyze effects of varying other parameters. Fixing the parameters at different values quantitatively changes the results. Therefore, we focus on overall behaviors that resist qualitative changes due to alternative choices. Perhaps most importantly, varying topology and the functional response control parameters profoundly affect persistence. Fig. 2a shows the effect of varying q and c on 30-species webs with an intermediate level (17) of C0 = 0.15 for food webs with initial topologies built using the random, cascade and niche models. All other input parameters are constant across all trials of the stochastic models unless otherwise indicated. Most or all species go extinct in every trial of random webs and q and c have little if any effect on their relative persistence (P < 0.05). The structural constraints provided by the cascade model and especially the niche model increase P by more than an order of magnitude. In addition to this enormous effect of network structure, a large change in persistence occurs when q is increased from 0 to 0.1. In this range of q, cascade-web P increases 32% from 0.34 to 0.44 and niche-web P increases 44% from 0.43 to 0.62. Compared to cascade webs, niche webs are 27% to 50% more robust for any fixed q from 0 to 0.3 and more strongly increase in persistence for q > 0. Fig. 2b shows that predator interference causes a similar change in the persistence of 30-species webs when c varies across a biologically reasonable (34) range. The effect of c on persistence is similar to the effects of q but, unlike q’s asymptotic effects, increasing c continually increases persistence across the whole range of values examined. Due to the similar effects of q and c, we present further results only for intermediately robust responses with q = 0.2 or c = 1.0, a choice that highlights the effects of altering other model parameters in a representative manner.

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S = 30, C = 0.15, Functional response = FH 0

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c (b) Figure 2. Mean overall persistence (P) of model food webs vs. functional response control parameter for networks built using the random (‘), cascade (□) and niche (○) models. In (a) q controls the parameterized Holling (type II to “type II.3”) functional response (Eq. 2); in (b) c controls the Beddington-DeAngelis (BD) predator interference functional response (Eq. 3). All networks initially have S0 = 30 and C0 = 0.15. Values shown are averages of 500 trials.

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Relative persistence (PR = PA / S0) of niche-model webs decreases linearly both with increasing initial network size (S0) and with increasing initial connectance (C0) (Fig. 3) as shown by linear regressions of PR as a function of the product S0C0, the network’s initial value of L/S. For the type II.2 response (q=0.2) with constant C0 = 0.15, PR = 0.87 – 0.05 S0 C0 (R2=0.48, n=2500); with constant S0 = 30, PR = 0.93 – 0.06 S0 C0 (R2=0.23, n=3500). Despite the negative effect of S0 on PR, absolute persistence (PA) increases with S0 from roughly 11 when S0 = 15 to approximately 25 when S0 = 50. We compared variation in CP with SP among persistent webs that were initially constructed with the niche model to two other sets of model webs (Fig. 4). These sets were created by starting with a set of niche webs using fixed parameters S0 = 30 and C0 = 0.15 and then randomly deleting species (41,42) to create networks with the same S as the persistent webs. Two deletion algorithms were used. One deletes species entirely at random and the other randomly deletes only non-basal species (42). C of niche webs increases with the number of entirely random deletions but varies little when basal species are protected. Despite the strong negative effects of C0 on P, CP of the most robust webs (SP > 21, PR > 0.7) is typically greater than the C of niche webs subjected to random deletions (Fig. 4). This suggests that structurally peculiar subsets of niche webs with relatively high C yield remarkably persistent networks (42). Both S and C affect many topological properties of empirical and niche-model webs (17,18,20,43). We examined how dynamic extinctions affect network topology by controlling for these effects and comparing the persistent webs with two sets of 1000 niche webs (Fig. 5a-e). One set had the initial values of S0 = 30 and C0 = 0.15 as inputs and non-basal species were randomly deleted until S = Sp. This compares persistent webs of a certain size to similarly sized niche webs subjected to randomized extinctions that leave C relatively unchanged (C ≈ C0 ≈ CP, Fig. 4). The second set was created using the values S = SP and C = CP as inputs into the niche model, allowing comparison between persistent webs of a certain size and similarly sized niche webs not subject to extinctions.

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0.8 Type III functional response, q = 0.2 BD functional response, c = 1

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(b) Figure 3. Mean overall persistence (P) of model food webs vs. (a) initial network size S0 for niche model networks with C0 = 0.15, and (b) initial network connectance C0 for niche model networks with S0 = 30. The dynamical model uses (○) the parameterized Holling type II.2 functional response where q = 0.2 (Eq. 2) and (□) the BeddingtonDeAngelis (BD) predator interference functional response with c = 1 (Eq. 3). Values shown are averages of 500 trials. The regression lines are (a) Type II.2: P = 0.874 – 0.00770S0, r 2 = 0.996; BD: P = 0.799 – 0.00682S0, r 2 = 0.992 and (b) Type II.2: P = 0.927 – 1.923C0, r 2 = 0.998; BD: P = 0.862 – 1.799C0, r 2 = 0.997.

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Figure 4. Mean connectance C of model food webs versus dynamically persistent network size S (×), with error bars showing plus and minus two standard errors of the estimated mean. The points without error bars show the mean connectance of 1000 niche model networks that have species deleted at random (○) or have consumer species deleted at random (□). All initial networks are built using the niche model with S0 = 30, C0 = 0.15, and the dynamical model uses our Holling “type II.2” functional response where q = 0.2 (Eq. 2).

Compared to either set of niche webs, persistent webs consistently have higher fraction of basal species and consumers with lower mean trophic levels, especially in larger most persistent webs (SP > 25, Fig. 5ab). Both of these properties vary with SP in the same direction but less strongly as the properties vary with S in niche webs. The fractions of consumer species that are omnivores or herbivores are higher in the persistent webs than in the niche webs (Fig. 5c-d). This helps explain persistent webs’ lower mean trophic levels. The differences in herbivore and basal species richness tend to lose their statistical significance as webs get smaller, while the differences in mean trophic level also get smaller but remain significant. The fraction of omnivorous consumers was often slightly (5-10%) though not significantly higher in the highly robust persistent webs (SP > 25), whereas there was a slight deficit of omnivores in less robust persistent networks (S P < 15). The standard

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4 3.8

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Figure 5. Mean and variation of model food-web properties versus persistent network size S (×). Error bars show plus and minus two standard errors of the estimated mean. Points without error bars show the mean property value in 1000 niche model networks with the same size and connectance as the dynamical model networks (○) and in 1000 niche model networks with the same initial size and connectance as the dynamically constrained networks that then had consumer species deleted at random (□). Properties shown are (a) fraction of basal species, (b) trophic level, (c) fraction of consumers that are omnivores, (d) fraction of consumers that are herbivores, and (e) standard deviation of node connectivity. Initial networks are built using the niche model with S0 = 30, C0 = 0.15, and the dynamical model uses our Holling “type II.2” functional response with q = 0.2 (Eq. 2).

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deviations of node connectivity were similar between persistent and niche webs but random deletions increased these deviations above those in persistent webs (Fig. 5e). This similarity also applies to the standard deviation of the number of incoming and outgoing links taken separately, properties previous termed the generality and vulnerability, respectively (17). We examined omnivory more finely by altering the skewness of omnivores’ preference for prey at different trophic levels. Such skewness has profound effects on overall persistence, P (Fig. 6), similar to the effects of varying the functional-response parameter q. Niche webs are most persistent (P ≈ 0.42 when q = 0 and P ≈ 0.64 when q = 0.2) when omnivores prefer lower trophic-level resources but avoid near exclusive consumption of the lowest trophic-level resources (0.2 < skewness < 0.8). Persistence drastically falls to as low as P ≈ 0.25 when q = 0 and P ≈ 0.34 when q = 0.2 as omnivores more strongly prefer upper trophiclevel resources (skewness = 10). 0.7

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Figure 6. Mean (n=500) overall persistence P of model food webs vs. skewness k of the prey preference of omnivores. When k = 1, all prey preferences of an omnivore are equal; when k < 1, low trophic level prey are preferred and when k > 1, high trophic level prey are preferred (see methods). All networks initially have S0 = 30, C0 = 0.15, and the dynamical model uses parameterized Holling type II (q = 0) and II.2 (q = 0.2) functional responses (Eq. 2).

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3.1. Effects of Structure on Dynamics

Our results generally illuminate how the structure of ecological networks may influence their function by examining the effects of diversity and complexity on in silico ecosystem dynamics. One early and remarkably durable theory based on linear stability analyses of random networks (6) proposed that S and C have hyperbolically negative effects on stability. Qualitatively similar effects occur in our nonlinear analyses of more ecologically realistic networks, but the effects are linear rather than hyperbolic (6), perhaps due to the differences between linear stability and nonlinear persistence. C affects persistence much more strongly than does S. This is illustrated by the regressions in which variance in C explains over twice as much variance of PR as does variance in S. This greater importance of C than S to persistence had been previously noted but the negative effects of C observed here are opposite the previously noted positive effects (42,10,2). Analyzing the effects of deleting species or otherwise challenging persistent webs to study their robustness may clarify this discrepancy. Beyond the classic effects of S and C on dynamics, our study illustrates the overriding importance of the overall arrangement of links among species. Random webs have almost no persistence while the hierarchal ordering of the cascade model vastly increases persistence. The contiguous niches and looping (43) in the niche model appears confer even more persistence on food-web networks. The hierarchical ordering of the cascade and niche models is easily interpreted as a mechanistic formalization of energy flowing from plants to upper trophic levels. Models that ignore such distinctions between plants and animals by making all species capable of growing without consuming other species (2) fail to detect the significance of nonrandom and hierarchical network structure (45). Niche space as formalized by the niche model is much less easily interpreted and deserves more study to understand which evolutionary, ecological, and mathematical factors underlie the improved the model’s improved empirical fit (17) and persistence (Fig. 2).

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These effects of network structure on dynamics closely mirror the degree to which model networks mimic the structure of ecological networks (17,18,20,46,47,21). The random model mimics very few properties of networks and dynamically sustains very few species. The cascade model mimics several natural network properties such as the fractions of top, intermediate and basal species and dynamically sustains ~50% or less of the original species within cascade webs. The niche model mimics over a dozen network properties and typically sustains between half and two thirds of the original species within niche webs. This suggests that a structural model that even more accurately mimics natural webs will dynamically sustain even larger fractions of species. This also suggests that scientists should be somewhat skeptical of models that mimic very few network properties, especially if these properties that can be reproduced by vast number of highly disparate models (48). Instead, models such as the niche model that matches a broad and well populated range of network properties may provide much more robust frameworks for integrating and exploring network structure and dynamics of the natural systems of interest. 3.2. Effects of Dynamics on Structure

Our work illuminates how the functioning of ecological networks influences their structure by examining the effects of nonlinear dynamics on the topology of complex food webs. Within network science, such analyses and influences may be only generalizable to networks such as food webs and pollination webs (45) whose nodes critically depend on interactions for their continued existence. Within ecology, our results show for the first time that the stabilizing effects of both predator interference and respective decelerated and accelerated feeding on rare and abundant resources found in small modules of two species also apply to much larger networks with 30 or more species. This enables large complex food webs to sustain many more species than networks governed by standard type II responses. This remarkable persistence greatly increases the potential to theoretically and computationally add

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other ecological processes such as facilitation, age-structured populations, migration, and environmental stochasticity to models of large ecological networks, which should further facilitate exploration of their effects on ecological structure and dynamics. We also show that small and perhaps empirically undetectable changes in functional responses foster greatly increased persistence in model ecosystems (21). This suggests that tiny amounts of prey switching behavior of consumers (37,2) or refuge seeking behavior of resources (36,49) has large effects on the structure and dynamics of complex ecological networks. This suggestion complements recent empirical findings (33,49), suggesting these functions as some of nature’s more prevalent and important stabilizing strategies. More effects of network function on network structure are seen in comparisons between persistent webs and webs generated by structural models free from explicit dynamics. Persistent webs typically have C similar to that in niche webs whose consumers are randomly deleted but lower than that in niche webs subjected to random deletions of any species. More strikingly, persistent webs have higher fractions of basal species and consumers with lower mean trophic levels than do niche webs. This is consistent with the niche model’s systematic overestimation of empirically observed food-chain lengths (17) assuming that empirical webs have more persistent topologies than do niche webs. While the SD of node connectivity shows few differences between niche webs subjected to dynamic loss of species and random loss of consumers, more detailed investigation of degree distributions (20) could illuminate differences hidden by our relatively coarse analysis. Given the niche model’s overestimation of the mean trophic level of consumers in large persistent webs by almost a whole level (Fig. 5a) and its underestimation of the fraction herbivores by ~0.07 (Fig. 5c), we tested the niche model against these properties of the seven empirical webs originally compared to the niche model (17). Table 1 shows that the niche model consistently overestimates mean trophic level by 0.2-2.4 levels and consistently underestimates the fraction of herbivores by

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0.01-0.32. Apparently, dynamics alters these properties of niche webs to become even more similar to empirically observed properties. The empirically observed fraction of basal species is well explained by the niche model (17), so the higher fraction of basal species observed in the dynamically constrained networks (Fig. 5b) appears to conflict with empirical findings. This discrepancy may be due to highly aggregated and poorly described basal species in the empirical data. For example, basal species in the St. Martin island food web (50) are categories of plant material such as seeds, leaves, etc. Many basal taxa in the Bridge Brook Lake (51) food web are trophically identical, suggesting that the trophic links are poorly resolved (52). Therefore, the fraction of basal species in the observed trophic-species networks and the niche model’s fit to these fractions could be methodological artifacts of taxonomic and trophic resolution. The importance of basal species to persistence emphasizes the need for high quality data resolved evenly at all trophic levels (53). Alternatively, artifacts of the dynamical model Table 1. Errors of niche model predictions of the fraction of herbivores (Herbivory) and mean trophic level (TL) of consumers in empirical food webs. S is the number of trophic species. C is directed connectance. Error is measured both as the difference between the model’s mean property and the empirically observed property (in parentheses) and in more rigorously comparable terms of the number of model standard deviations that the empirically observed property differs from the model’s mean (17).

Food Web Name

S

C

Herbivory

TL of consumers

St. Martin Island

42

0.12

-2.7 (-0.15)

1.4 (0.79)

Bridge Brook L.

25

0.17

-3.9 (-0.19)

1.5 (1.23)

Coachella Valley

29

0.31

-1.3 (-0.04)

0.6 (1.24)

Chesapeake Bay

31

0.072

-0.2 (-0.01)

0.6 (0.21)

Skipwith Pond

25

0.32

-7.8 (-0.29)

0.1 (2.39)

Ythan Estuary

78

0.061

-4.1 (-0.20)

1.6 (0.60)

Little Rock L.

92

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-12.7 (-0.32)

2.5 (1.52)

Mean

-4.62 (-0.17)

1.17 (1.14)

Std error

1.65 (0.04)

0.30 (0.27)

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might cause the discrepancy (45). Our models assume that basal species do not compete for shared resources. Adding competition among basal species might lower the fraction of basal species in the persistent webs. 3.3. Omnivory

One of the more confusing interdependencies between food-web structure and dynamics concerns the issue of omnivory. There is a close positive and confounding relationship between omnivory and C in earlier studies, (8,10) since increasing C typically makes consumers more omnivorous and increasing omnivory typically increases C. We help clarify this issue by controlling for the strong effects of C on persistence (Fig. 4) and showing that the prevalence of omnivorous consumers in persistent webs is usually similar to that in niche webs (Fig. 5d), which is typically much less than in cascade webs (17). If structural omnivory had an unusually strong positive effect on persistence, one would expect higher omnivory in the most persistent niche webs and more persistence in cascade webs. This is not generally supported by our results. Contemporary modeling studies also tend to confound increasing omnivory with lowering consumers’ trophic levels by increasing omnivory in a narrow fashion. That is, omnivory is typically created by adding short paths that enable carnivores to consume primary production (8,10). Adding this type of omnivory lowers the consumer’s trophic level. Omnivory that increases a consumer’s trophic level, for example, by adding a carnivorous links to an herbivore’s diet, is typically avoided. Omnivores that prefer higher trophic level prey strongly decrease persistence compared to omnivores lacking such preference, while variable preference for low levels has much less effect (Fig. 6). These findings, combined with consumers’ relatively low trophic levels and high prevalence of basal species and herbivores in the most persistent niche webs, suggest that shortening food chains and reducing trophic levels account for the stabilizing effects previously attributed to omnivory. In contrast, omnivory strongly decreases persistence in food webs when omnivores engage in the empirically unusual (19) destabilizing behavior of preferring prey at higher trophic levels.

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4. Conclusion

Our analyses address several historically perplexing aspects of the remarkable complexity and persistence of natural ecosystems and show how more empirically prevalent aspects of ecological interactions (17,33,25,49) may confer persistence on large complex ecosystems. Both food-web structure as characterized by the relatively successful the niche model and food-web function as characterized by decelerated consumption of rare resources (49), predator interference (33), and omnivores’ preferences for lower trophic-level prey (25) greatly increase the diversity and complexity that persists in ecological networks. While all models are simplifications of nature, formal inclusion of these frequently observed regularities indicate that our ecological models may be the most biologically informed and empirically well-corroborated in terms of their detailed diversity, complexity, structure, function, and dynamics. Some of the increased persistence resulting from including these factors appears to have been mistakenly attributed to unqualified omnivory. The strong effects of predator interference and decelerated and accelerated feeding on relatively rare and abundant resources, respectively, suggests that other behaviors that reduce consumption of rare resources, e.g. prey switching (37,2), will also stabilize large complex networks. In contrast, responses that increase consumption of rarer and higher trophic level resources, e.g. economic exploitation of relatively rare carnivorous fishes (55), can be expected to decrease persistence. Perhaps a larger lesson to be taught by these studies is that several solutions to the devilishly difficult problem of understanding the structure and nonlinear function of complex networks have been found in the subtle details of these networks. That is, the fine structure of the particular location of the links may matter much more than particular distributions of degrees among nodes or species among trophic levels that can be simulated by a wide variety of network models (21). Similarly, the particular trajectory by which functional responses reach their maximum consumption rates may matter more than the presence or magnitude of these rates’ asymptotic maxima. While interdisciplinary

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network theory has much to offer many scientific disciplines (56,55), useful application of such theory may critically depend on understanding the devil of disciplinary details essential to the structure and function of real-world complex networks. Within the discipline of ecology, the persistent models described here provide new tools to explore non-trophic processes including invasions, extinctions, experimental manipulations, environmental variability, and spatial processes. Such processes could be simulated by manipulating our model’s parameters e.g. stochastically varying basal species’ carrying capacity, and adding different functions e.g. densitydependent migration. More study of these models as well as empirical and especially experimental tests of their findings could significantly refine our results. Such combined studies could do much to bring about exciting new insights regarding the trophic and non-trophic interactions in the large complex networks that sustain the stunning, yet tragically diminishing, levels of diversity in nature. Acknowledgements

Jennifer A. Dunne, Ulrich Brose, and Jessica Green are greatly appreciated for comments on the manuscript. NSF provided support for RJW and NDM. NDM especially thanks the National Center for Ecological Analysis and Synthesis and the NSF funded IGERT Program in Nonlinear Systems and hospitality of the Telluride House, both at Cornell University, for support. Correspondence and Requests for materials should be addressed to: E-mail: [email protected], Telephone: 510-295-7624, Fax: 970-349-7481 References 1. 2. 3. 4.

McCann, K. (2000). The diversity-stability debate. Nature. 405, 228-233. Kondoh, M. (2003). Foraging adaptation and the relationship between food-web complexity and stability. Science. 299, 1388-1391. Paine, R.T. (1966). Food web complexity and species diversity. American Naturalist. 100, 65-75. MacArthur, R.H. (1955). Fluctuation of animal populations and a measure of community stability. Ecology. 36, 533-536.

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CHAPTER 13 INFECTION TRANSMISSION THROUGH NETWORKS

James S. Koopman Dept. of Epidemiology, University of Michigan, USA [email protected]

1. Introduction The epidemiologic analysis of systems that cause infection in human populations inform the spending of billions of dollars and affect the lives of millions. This is the case for deciding on how to stem the ravages of a new pandemic flu strain, how to treat water to prevent transmission of diverse infectious agents such as Cryptosporidia, how to focus HIV treatment to stop transmission, and how to respond to emerging infections like SARS. Network models are making new and important contributions to such analyses. Network model abstractions, however, may not always be appropriate. Infection transmission systems are complex and diverse. Infectious agents usually spread from one individual, species, or environment to another via multiple modes. For example many respiratory pathogens can spread both through the air over long distances, via droplets from sneezing over short distances, and via touching mucosa then surfaces that are touched by others who touch their mucosa. Each mode alone would generate different transmission dynamics. Combinations of modes might occur in patterns that make one or another mode the key to control. For example even though droplet spread may be more frequent than airborne transmission, airborne transmission might play a key role in the network and thus offer a better target for control. The effect of interventions on 449

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different modes of transmission varies by the (a) paths that infection transmission can traverse, (b) natural histories of infection and immunity, (c) survivability of infectious agents outside a host, (d) environmental factors such as temperature, humidity, acidity, oxidative potential, etc., (e) the dose of infectious agent required to initiate an infection, (f) occurrence of other comorbidities affecting host immunocompetence, (g) complex evolutionary patterns that generate diverse strains with only partial cross-reactivity, and many other factors. For more than a century, the major traditions for making simplifying model assumptions were based on differential equation analyses that ignored the influence of enduring contact networks. Those traditions arose to accommodate the analytical tools available rather than as the result of a careful consideration of what simplifying assumptions might do to the inferences made from analyzing a model. Those traditions provided insights and helped organize thinking about infection control. But, they failed to generate a progressive science of transmission systems characterized by increasingly robust and data-driven model development. Infection transmission science is still a data-poor discipline with many isolated methods and theories. Perhaps network analyses could play an integrative role that helps epidemiologists validate their methods and theories. One reason for the failure of differential equation models to launch a transmission system science is that they intrinsically ignore a number of network phenomena that are centrally importannt to infection transmission. Newman lists some of these (1,2) including (a) the small world effect, (b) transitivity, (c) degree distributions, (d) network resilience, (e) mixing patterns, (f) degree correlations, (h) community structure, (i) network navigation, (j) component size distributions, and (k) distributions of betweenness centrality (see Chapters 1 and 2). Powerful new tools for analyzing infection transmission networks have been developed. Beginning with the specification of probability generating functions (pgfs) for network degree distributions, a series of analytic methods have made it possible to solve for the probabilities of epidemics and the expected sizes of epidemics under increasingly complex conditions without burdensome simulations that make it difficult to assess the implications of model variations (2-12).

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Additionally, efficient simulation and network analysis algorithms, and greater computational power improve our ability to analyze networks currently intractable by theory alone. Nevertheless, network analyses using these new tools employ simplifying assumptions that are as extreme as the assumption of mass action in differential equation models. If the effects of these radically unrealistic assumptions in network models are ignored by network scientists, then network model analyses are likely to leave transmission system science in the state of very slow growth generated by mass action models. One reason to hope that network analyses could lead to rapid growth in transmission system science is their potential to relate to new sources of data, including data on contact patterns, data on environmental agent identification, and data on patterns of nucleotide sequence variation in infected individuals. For that promise to be realized, epidemiologists have to get involved with network scientists in the work of analyzing infection transmission. Otherwise the needed data will not materialize and the relevant questions that can advance both the theory and practice of infection control will not be addressed. To involve epidemiologists more fully, network scientists will do well to address the identification and control of risk factors, an organizing principle in epidemiologic analysis. Accordingly, I urge network scientists to pursue these three directions in the analysis of infection transmission: •

•

•

Establishing a method for comparing inferences derived from network models with those derived from other model forms. This will improve the quality and efficiency of infection control decisions and motivate better data collection. Create a theoretical framework to integrate nucleotide sequence data from clinical and environmental isolates into transmission system models. Those data hold a large and untapped potential to increase the accuracy of network model specification. Consider the role of the distributions of risk factors within a network in determining its epidemic potential. Developing this lexicon of risk factors in network science will further align the

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interests of the epidemiologic and network sciences. Only when network theorists get into this dominant mode of epidemiologic thinking are epidemiologists likely to begin using and developing their work. After further introducing these three issues, I will review broad issues in network analysis of infection transmission systems. Rather than thoroughly reviewing all recent advances, I seek to provide a vision of tasks for network scientists that I think will contribute most to infection control. An area I do not cover where network analysis is particularly important is the evolution of infectious diseases (13). 2. Inference Robustness Assessment The robustness of infection control decisions or causal inferences is assessed by relaxing the simplifying assumptions of a model and evaluating whether that leads to a change in the inference. Note that we are talking here about assessing the robustness of inferences, not of models. We are not talking about the robustness of statistical inference. Statistical inferences are relevant only from samples to statistical target populations and do not depend upon causal inferences. The inferences we are concerned with are about the general applicability of a theory or about the causal consequences of control actions. The simplifying assumptions of a model may be relaxed by changing model elements while keeping the form of the model constant. For example a network model may have a process that generates a network structure. Some assumption in that process could be relaxed by elaborating the process so it is more realistic. Certain assumptions are intrinsic to model forms. The intrinsic assumptions of network models and of compartmental models using mass action contact formulations are both important to relax. One can relax the assumptions of network models by transiting to mass action formulated compartmental models and vice versa. Another process of inference robustness assessment is to transit from one model form making one set of simplifying assumptions to another model form making a different set of model assumptions (14-16). One

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ideal for such transitions is to formulate transitions between models in such a way that both model forms should theoretically generate identical output. This can be done for example in the case of the mass action network model transition by formulating dynamic network models (16). Formulating models of different types such that both should theoretically behave the same has two advantages (14-16). First, it helps validate model code by insuring that two quite different model forms generate the same output when they theoretically should do so. Second, it allows for relaxation of assumptions within a single model form rather than across two model forms where all the different assumptions may not be identified. Inference robustness assessment validates a model for a specific use. Validation of any model for all purposes is impossible. Demonstrating that a model generates observed patterns does not validate any model use. Conversely, a model can be validated for a specific use even if many aspects of the patterns it generates differ from observed patterns. A model based inference is not invalidated just because some aspects of the model differ from those of the real world. It is only invalidated if realistically relaxing assumptions changes inferences. Thus inference robustness assessment entails demonstrating that the ways a model differs from the real world do not affect model based inferences. Given the multitude of ways that simplifying model assumptions might be realistically relaxed, the task of inference robustness assessment can never come to a definitive end. It is practically impossible to look at every combination of recognized ways to relax simplifying model assumptions in order to assess inference robustness. Even if all recognized ways to relax model assumptions could be examined, insightful scientists would discover new assumptions and ways to relax them. The best we can hope for from inference robustness assessment is a process that leads to consensus. Inference robustness assessment is not a formula for finding the truth. It is a social process that can bring out threats to finding the truth that individuals pursuing the truth on their own may not perceive.

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2.1. Transitions within Mass Action Models from Continuous Population Deterministic to Discrete Individual Stochastic Models We deal first with this deterministic to stochastic transition within mass action models even though it does not involve network models because the issues addressed in this transition are part of transitions from ordinary differential equation models of infection transmission to network models. But these issues are better clarified without adding the network issues. Mass action models can be formulated either as deterministic continuous population models using differential equations or stochastic discrete individual models. The continuous population deterministic models assume that the size of each compartment in the model is effectively infinite. The stochastic compartmental models relax that assumption but in so doing they must add a stochastic process to the model and assumptions about that stochastic process. Usually the stochastic model is the more realistic one. It makes less radical and more realistic assumptions than the deterministic model. It may have more parameters because it adds realism to the model, but that does not mean it makes more assumptions – just that it makes more realistic assumptions. In an illustrative case, an inference about the relative utility of contact tracing with quarantine programs and mass immunization was made using deterministic mass action models (17,18). These models had nice characteristics such as the inclusion of the human resources needed to carry out these programs in the model. They were analyzed in a manner that demonstrated the robustness of the inference regarding the relative utility of these two programs to a variety of assumptions about the nature of the transmission system. But this inference, in turn, was shown not to be robust to the deterministic model assumptions of continuous populations (19-21). Various reasons for the lack of robustness were postulated (20) with local die out of infection that occurs in discrete individual stochastic models but not in continuous population deterministic models (22) being the main one. Stochastic effects like those just discussed for smallpox are not always important. An illustration of how the natural history of infection and immunity may determine whether stochastic phenomena must be

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captured is found in Rohani et al. (23). They find that the dynamics of whooping cough are strongly determined by stochastic phenomena but measles is not. The transition from deterministic mass action models formulated as ordinary differential equations (ODEs) to stochastic mass action models creates a dramatic change in system equilibrium states for infection transmission models (24). Whereas the deterministic ODE models may have mathematically derivable equilibrium relationships, stochastic models always have a chance of going to the state of no one being infected and thus reach an equilibrium where no one is infected. There is, however, a pseudo-equilibrium for stochastic infection transmission models that given large population sizes approximates the equilibrium of the deterministic models. It is the deviations from this equality that are informative. Given limited population sizes, when endemic infection levels bounce up by chance, the number of recoveries driving the infection levels down always increases because these are proportional to the number of infected individuals. In contrast when infection levels bounce down due to chance, the force driving them up (infected cases that are the source of transmissions) always decreases (25,26). This drives equilibrium levels down. Likewise as population structure increases by the specification of further mixing groups (22), the chance that infection will die out of a population subgroup increases and this further lowers the equilibrium level of infection in the stochastic mass action model. Network models with individuals as nodes can be viewed in this regard as structured mass action models where every linkage is a two person mixing group. Thus when transiting from deterministic mass action ODE models to network models, the stochastic phenomena just discussed will be especially important. But there are even more important aspects of this transition which we consider next. 2.2. Stochastic Compartmental Mass Action — Network Transitions Consider a population with individuals always in motion who establish transient linkages that can transmit infection. Linkages may consist of joint presence in a building, the use of a common bathroom,

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conversations, hugs, preparing and sharing food, etc. Some of these transient relationships recur regularly over many years such as those in families. Others recur only over months, such as those in academic institutions. Some recur only a couple times, such as those involved in business deals. And some never recur. The temporal pattern of recurrence may be quite different for different settings. Most social encounters are temporally structured so that they occur at regular intervals or at least after minimal separation times. Now contrast the mass action and network abstractions of such a population. One network model abstraction has individuals as nodes with directed or undirected arcs that are fixed by some definition of what transient relationships are sufficiently strong and long enough or will come into play at crucial moments so as to constitute an arc. It assumes individuals do not move. The contact process generating transmission events across an arc is almost always assumed to be Poisson with exponentially distributed event intervals. These assumptions about fixed relationships and no movement and Poisson processes for events may be true for models of some physical systems, but are not remotely true for any infection transmission system. The mass action model, in contrast, assumes all interactions are instantaneous with instantaneously thorough mixing that results from extremely rapid movement after each contact and accumulate numbers of contacts in a Poisson process (27). These assumptions may also be true for models of some physical systems, but again, are not remotely true for any infection transmission system. Both model forms making such radically unrealistic assumptions may be useful. But they might lead to different inferences. The question then is which assumptions are affecting the robustness of inferences. Are the network model inferences or the mass action model inferences more robust? Network and mass action models make almost opposite unrealistic assumptions about both movement of individuals and duration of contact. Both make assumptions about the timing of contacts that are unrealistic and have been shown recently to have important influences on transmission system dynamics and inferences about the effectiveness of control measures (28). Assumptions about movement and duration of contact are both more easily relaxed in compartmental models. This can

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be done by defining contact groups and movement of individuals between contact groups. But assumptions about micro-network structure are more easily relaxed in network models. To see what we mean by this, consider a transition between model forms. A network model equivalent to the mass action model could be formulated with linkages between every pair of individuals and a rate of transmission corresponding to the effective contact rate between any pair of individuals in the mass action model. The network model can then relax the mass action assumptions of instantaneous contact and instantaneously thorough mixing by eliminating a fraction of arcs and correspondingly increasing transmission probabilities to keep constant the overall rate of contact events. As the nodal degree is decreased and the transmission probabilities across edges are increased, the opportunities for infection transmission are decreased. Fewer contacts of infected individuals are with susceptible individuals because of the increased chance that a potential contact event is with the source case for transmission. The transitions from mass action to the equivalent network model and then the relaxation of mass action assumptions in the network model form could and should be used to assess the robustness of any inferences in the mass action model to realistic violation of the instantaneous contact assumption. Alternatively the transition from network to mass action models could be formulated by having a rate of formation and break up for pairs making the network model a dynamic network model. If the formation of pairs results from a mass action process, then as the breakup rates are increased, the network model approximates a mass action model (16,29). Even in the completely identical random network and mass action formulations, the network model has patterns of arcs in small sets of individuals that the mass action model does not have. We refer to such patterns involving triads, tetrads, etc. as micro-network structure. This micro-network structure consists of patterns of relationships between two contacts of an individual, or between second or higher generations of contacts of an individual. For compartmental models to handle such micro-network structures, they would have to define mixing groups for every small group conformation (14,15,30). That would so quickly explode model structure that it would make the model useless. The

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micro-network structures in network models can be elaborated in ways that are impossible to elaborate within the context of the mass action model. Micro-network structures may derive from basic social processes and thus be scientifically generalizable across populations. Network models can relax the random mixing assumptions of the mass action model by assuming different micro-network patterns or by formulating forces that lead to different patterns. In one such exercise, micro-network structure was changed by adding clustering of contacts (31). This led to notable effects on the probability of extinction in stochastic models and the fraction of the population left susceptible at the end of an epidemic. In another such exercise, effects on the equilibrium level of infection for infections without permanent immunity were found to dramatically change across the progressive change of formation and breakup rates (16). Such a transition was further explored in limited pair formation models with similar inferences being made about the effects of changing pair formation and breakup rates in relationship to infection recovery rates (32). The behavior and formulation in terms of model parameters of key epidemiological constants such as the basic reproduction number, endemic infection levels, and critical vaccination levels is significantly different for mass action and network models even when both microand macro-network structure is random. This has been shown using probability generating function formulations of networks (8) by analyzing pair models (33), and by analyzing dynamic network models (16). Any aspect of contact structure that can be readily defined within the context of compartmental models by specifying risk groups and mixing groups we define as macro-network structure. Macro-network patterns can be defined in either the compartmental or network model context by formulating rates of contact between different classes of individuals. Each real world situation modeled is likely to have a unique macronetwork structure with many hypotheses about the nature and effects of that structure that deserve exploration. That means macro-network structure is not as generalizable as micro-network structure.

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Although both mass action and network models can relax macronetwork assumptions, they do not do so in an identical fashion. Network models assign limited numbers of ongoing linkages between individuals at different sites in order to create the macro-network structure while mass action models create homogeneous contact processes at a site. That is to say, network models incorporate micro-network structures into their definition of macro-network structures while mass action models do not. Model analyses of highly structured continuous populations using mass action assumptions are recognizing the need to assess the robustness of their inferences by considering network models (34). Network model analyses need to do the reverse. Fixed network models assume no one is changing their spatial or social relationships to other individuals. There is no motion on these dimensions. Mass action models assume everyone is in so much motion that mixing is instantaneously thorough. Currently, mass action models can more flexibly relax their macro-network motion assumptions than network models can. Movement assumptions can be relaxed in the compartmental model form by defining compartments of individuals in different contexts and moving individuals between contexts (35-37). Movement is not explicit in network models. Assumptions about movement effects on symmetries of contact, however, can be addressed in network models (5). Moreover work on integrating movement into network model analysis is proceeding (38). A common compartmental model of structured populations called structured mixing (30) makes assumptions that obscure movement effects between different sites. That formulation takes a statistical mechanics approach to movement so that it assumes all individuals are simultaneously at all sites at all times with specified probabilities. Thus one does not explicitly formulate movement. Metapopulation models (35-37,39) and mechanistic movement models (40) formulate movement more explicitly. Mechanistic models are more detailed and keep track of everyone at any site by their population of origin while metapopulation models keep less track of the history of where individuals have been (40). Besides greater flexibility for relaxing motion assumptions, there are other reasons to transit from network models to mass action models. One

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is that analysis of the effects of relaxing macro-network assumptions may be computationally easier. Another is that when models are used in data analyses, the number of parameters to be estimated from the data must be reduced and that might be easier using mass action model forms. Later in Section 2.5 on mathematical analysis of network models we will suggest that advances in network analysis might make network models more flexible for both these situations. We have talked about robustness assessment so far as a process of elaborating models to relax simplifying assumptions and noted that network and mass action models have different advantages and disadvantages for this purpose. Inference robustness assessment, however, should also encompass the process of making inferences about model form or model parameters from data analysis. Different assumptions can be made in analytical models. But to undertake any data analysis, it may be necessary to reduce model parameters and perhaps even model structure. Thus a major reason for model transition in inference robustness assessment is to assist in the estimation of model parameters from data by imposing unrealistic simplifying assumptions. Network and mass action models do this in different ways and thus may offer robustness assessment for statistical inferences as well as for scientific inferences. Since statistical inferences are often used in making causal or control action inferences, inference robustness assessments should most often incorporate assessing the robustness of statistical inferences. 2.3. Dynamic Network Models Because both network and mass action models are far from reality, a robustness assessment might employ models where linkages between individuals form and break up. The simplest such models have only isolated pairs of individuals with no individual in two pairs at once. The relationship between pair models with these characteristics and mass action compartmental models has been explored in several studies (42). More general dynamic network models where pairing is a mass action process have enabled further exploration of the relationships between network and mass action models (16,29).

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The equilibrium relationships of dynamic pairing models that do not allow for concurrency can be derived by examining expected equilibrium status at pair formation and pair breakup (16). But equilibrium and other analyses for more general dynamic network models with continually varying nodal degree is a challenge that network scientists should consider taking on because that would provide a key tool for inference robustness assessments. 2.4. ODE Network Models of Correlation in Infection Status The transition from mass action deterministic models to network models may take another direction where pairings are added as separate compartments within an ODE model. An early influential model of this type was formulated by Dietz and Hadeler (43). Chick et al. explored the relationship between these deterministic models and stochastic dynamic network models (16). A generalization of the Dietz and Hadeler model stays in the deterministic ODE model framework but can examine additional model structure while still switching from instantaneous contact with instantaneous thorough mixing to fixed linkages (33,35,44-47). These are deterministic ODE models of infection spread in networks with either random structure or specifications on the level of increased linkage within small groups like triads (48). They have continuous populations rather than individuals but the continuous population is formulated in terms of pairs, triplets, or higher order structures. In these models, all relationships between individuals are enduring in nature. Assumptions about how infection status is distributed in higher order structures beyond pairs (triplets, tetrads, etc.), however, usually imply that these higher order structures are continually changing. Thus these network models differ significantly from fixed network models. Correlation models as presented by Rand (33) formulate the infection and immunity status of each egocentric linkage made by each individual. A random graph can be assumed. Triad closure probabilities can be specified (47). And even higher order structures may be formulated (48). Consider first the model formulation where the infection status of each egocentric linkage of each individual is formulated. These are called pair

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models. The basic approach is to construct models at the levels of individuals and dyads by first formulating models at the level of individuals, dyads, and triads and then formulating the triads in terms of individual infection status and dyad status. Many different assumptions can be made in the process of modeling triads in terms of individuals and dyads. Even with the random graph formulation, the differences in the mathematical relationships defining the basic reproduction number R0 in the mass action and correlation models are considerable (33). R0 represents the endemic or epidemic threshold parameter and under certain restrictive assumptions of random contact represents the number of secondary cases that an infected individual will generate over the entire course of their infection (49). There are also important differences in mathematical relationships for the endemic level of infection and for critical vaccination levels needed to control transmission. There are also differences in the dynamics of infection as correlation models broaden out the shape of an epidemic (31). Pair correlation models might be a good a first step in robustness assessments for inferences made using mass action assumptions. They relax those assumptions in the direction of network model assumptions but can preserve both random mixing assumptions and a degree of tractability for numerical analysis. After a first transition from mass action models to pair correlation models on a random graph, a second step would be to stay within the random graph context but pursue different models of correlation in higher order structures. To accomplish this, one can move to modeling quadruplets in terms of individuals, pairs and triplets, modeling quintuplets in terms of all the lower level structures, etc. One approach to such modeling has been “momentclosure” methods (33,44,47). But other approaches sometimes work better (46). As a second step in inference robustness assessment, one can proceed to add micro-network structure by specifying the probabilities of closed triads and of higher level structures (47). As a third step in relaxing model assumptions for an inference robustness assessment, different types of linkages from the same individual can be specified and the frequency of one type of linkage can be modeled as conditional on

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another (48). Using this approach for sexually transmitted infections (STIs), it has been clearly demonstrated that constraining formulations to the pair level is inadequate to capture important aspects of infection transmission dynamics (45). Since sexual linkages can be characterized fairly well into short and longer duration classes, and since there is a fair amount of data on these classes of linkages, this approach to adding complexity to micro-network structure is logical for STIs. As one moves from random graph correlation models to higher level structure models and as one moves from modeling correlations at the dyad level to correlations in these higher level structures, model complexity explodes and the logical structure becomes challenging, at least for a medical doctor epidemiologist. As one adds further population structure that puts individuals into different risk groups and further distinctions by type of interaction between individuals, again complexity explodes. Adding macro-network structure creates even further challenges but it is possible to formulate pair correlation models with metapopulation structure (48). But that state of affairs also holds for the “pgf” approach to network model formulation and analysis (8) which we deal with next. 2.5. Mathematical Analysis of Network Models with other Structures One aspect of network models that makes them particularly useful within the context of inference robustness assessment is the availability of percolation theory for their analysis and the success of applying percolation theory in models formulated using probability generating functions (pgf) of degree distributions and group structure or mixing site structure (8). As mentioned earlier, discrete individual mass action models with realistic population structure are mostly analyzed using simulations. The process of relaxing single assumptions and analyzing the effects of such relaxations using purely simulations is exceedingly laborious to the point that it will be undertaken only superficially. The potential for percolation analyses to provide solutions for complex stochastic models could significantly facilitate many inference robustness assessments. These solutions may be closed form solutions or solutions that only require simple numeric approximation to the solution

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of transcendental equations (4,8,9). Such solutions will be most readily available for simple model structures. As mentioned earlier, relaxing assumptions within simple models can provide some practical level of assurance about the robustness of inferences to these same assumptions in more complex models. Of course, relaxing assumptions two at a time may reveal lack of robustness in inferences that are not revealed by relaxing assumptions one at a time. But the conceptual clarity of how relaxing assumptions can demonstrate the lack of inference robustness that comes from simple models may be preferable within the social process of science. Many different lattice structures, structuring of local and distant contacts, degree distributions, and linkage patterns between degree distributions can be formulated. These include small world networks (50-55), scale-free network (56-58) different formulations of clustering (11,59,60), and other diverse specific structures (2,10,60-69). This proliferation of analyzable model structures is most encouraging for the development of robustness assessment methodologies. This is the case even when specific formulations are not very realistic. If two different formulations cover the extremes of reality and an inference holds across both of them, robustness to the characteristics on which they and presumably more realistic models differ has been demonstrated. A nice illustration of this principle was used in assessing inferences about SARS. Ancel Meyers et al. used Poisson and power law models to cover the extremes of more realistic models (5). Additionally it is easy to see how small world network formulations relate to realistic contact patterns and how assumptions in lattice models can be relaxed by adding distant linkages. Likewise it is easy to see how adding local structure to random graph models can realistically relax random graph assumptions to the point where small world formulations are achieved. A particularly promising model formulation using pgfs in solvable models formulates the network as a bipartite structure between humans sites where transmission takes place (4). Such models facilitate both adding realistic structure and the process of finding parameter values that fit observations. In a model of Mycoplasma pneumoniae transmission in a psychiatric hospital, Ancel Meyers, and Newman et al. made an inference that even though caregivers were at less risk than patients,

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restricting the number of wards on which they worked and protecting them from infection within the wards where they worked was more important to infection control that reducing patient to patient transmission. The robustness of this inference, however, to realistic relaxation of the simplifying assumptions in the model is in further need of assessment. The particular model form used makes the assumption that the number of infected patients on a ward does not affect the risk of a caregiver getting infected when they work on a ward. Realistic relaxation of this assumption seems likely to increase the utility of controlling patient to patient transmission while decreasing the utility of protecting caregivers from infection. While simulations relaxed other model assumptions such as the form of probability distributions, they did not relax this most important assumption. But then as we have said, all robustness assessments are in a state of incompletion and the social process of submitting analyses to scrutiny by other scientists will always reveal further possible robustness assessments. The fact that a clear formulation of network structure allowed for individuals other than the authors to assess assumptions whose realistic violation might be a threat to inference robustness makes this approach promising even though the particular inferences in this paper may not hold up. 2.6. Transiting from Undirected to Directed Graphs Most compartmental models of infection transmission assume that transmission is symmetric. This corresponds to arcs representing the potential for transmission between individuals being undirected. There are some strong differences in transmission potential in different directions between individuals with different characteristics. For example, sexual transmission often has higher transmission probabilities from males to females than from females to males because males deposit infectious agents into sites where females retain them. Another example is that transmission of nosocomial agents may be higher from doctor to patient than the other way around due to susceptibility differences. Not only is average transmissibility different, but the distribution of transmissibility may be different. Contagiousness, for example, might be

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more highly clustered than susceptibility. We will say more on that in Section 5.2 on modes of transmission later. In compartmental models, these asymmetries are usually handled by susceptibility and contagiousness parameters being assigned to individuals with different characteristics. It would seem that the same tactic would work for network models. But some asymmetries are total. If a transmission media is contaminated by one person and then picked up by another, timing and movement issues are likely to make at least some transmissions unidirectional. To my knowledge issues regarding mixed directionality of transmission have not been addressed in the compartmental model context. It would seem possible to do so by specifying a site compartment making bipartite graphs like those just discussed in the network models of Meyers et al. (4). But directionality is not as natural an issue in the compartmental model context as it is in the network model context. In individual level models, where transmission occurs via contamination of media and movement of individuals is explicitly modeled across time, directionality of transmission is an intrinsic part of the model (70). Pourbohloul et al. have shown that percolation analysis of directed graphs illustrates significantly different behavior than similar analysis of undirected graphs (3). They showed that in network models with mixed undirected and directed arcs the probability of large epidemics and the sizes of those epidemics are not identical as is the case for undirected graphs. A particular source of directionality addressed in these network models arises when patients go to make contact with health care workers when they become ill but the reverse does not happen. But directionality arising from different frequencies or timing of contamination of media events and uptake of contamination from media seem to be a much more ubiquitous source of asymmetry. Network models may provide the most feasible and flexible way to assess the robustness of inferences to assumptions about symmetry of contact. If the fraction of arcs that are directed can be varied by group, then the consequences of directed media contamination can be assessed and inference robustness limits demonstrated without having to model the media explicitly.

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2.7. Models Involving Contact Processes That Generate Networks Another way to pursue needed realism when inferences are not robust to the extreme assumptions of network or mass action models is to formulate models of contact process mechanisms. These may involve either modeling the media through which transmission takes place or specifying locations where contact might occur. Such models play an important role in bioterrorist and emerging infection control models that we will discuss in this section. Three traditions pursuing more realistic complexity in contact processes within infection transmission system models are “agent based” models, transportation based generation of contact models that handle movement in a metapopulation manner, and structured mixing models taking the statistical mechanics approach to movement. These three approaches are represented by the first three infection transmission modeling groups supported by NIH under the MIDAS (Modeling Infectious Disease Agent Systems) program (71). This is the major effort supported by the US government to understand how to control the population transmission dynamics of emerging infections and bioterrorist dissemination of infectious agents. The initial focus of these research groups has been on smallpox and pandemic influenza. A group linking Johns Hopkins University and the Brookings Institute has taken what they call an “agent-based” approach (72) which moves individuals around on realistically structured grids and defines contact by grid proximity. A group at Emory uses the structured mixing with the statistical mechanics approach discussed earlier (73,74). These are the same sort of models used to analyze smallpox that we mentioned earlier (21). A group now at Virginia Tech and formerly at Los Alamos National Labs builds simulations on models of the transportation network and generates contact by having infected individuals contaminate environments and susceptible individuals pick up contamination from environments (70,75). They call their model structure “EPISIMS”. All three of these models could be used by network analysts to relax simplifying assumptions in their network models and assess the robustness of inferences to realistic violation of their network

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assumptions. Conversely, all three of these model forms are very complex and do not readily allow for the generation of generalizable knowledge. Network analyses can relate to these model forms and improve their capacity to generate generalizable knowledge in two different ways. First, network models can be used within inference robustness assessment strategies that whittle down their complexity while both maintaining their accuracy by preserving key assumptions. Second, the output of these models includes highly dynamic and complex networks that should be described efficiently to better understand what is determining the behavior of these models. All three approaches could potentially use media contamination mechanisms to effect contact and transmission but only the models of the Virginia Tech/LANL group do so. This group is pursuring methods to directly formulate static networks for analysis. They first model stochastic processes that incorporate various forces leading to contact. They do so by modeling the media of transmission or by modeling contact processes at sites where contact is made. Modeling the media is especially applicable to agents that survive for some time in the environment. That includes a great many agents of interest for enteric transmission like noroviruses and rotaviruses as well as agents like smallpox. Even influenza is probably mostly spread through agents that have survived for at least hours in their transit between hosts. Each of the three MIDAS groups view and analyze the networks that are generated by their mechanistic simulations. The Virginia Tech and Emory groups view the resulting networks as bipartite graphs where humans link sites. The networks generated are subject to analyses regarding their structure. The purpose of analyzing networks generated by models is of course not to make inferences about the mechanisms that generated the network. These are completely known. Rather the purpose is to describe the networks along many dimensions and relate the behavior of the system to these descriptions. For example observed distributions of reproduction numbers for individuals in different parts of the network or with different risk factors can be described and related to epidemic size or epidemic risk. Analysis of networks generated by specific mechanistic processes and specific rules for defining network links may be one of the most

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important uses for network analyses. It could and has generated insights not possible without the network analysis (70). Complex contact process models make it difficult to determine what is really happening within a model. Network definition as the result of model processes and analysis of resulting network patterns can help perceive and analyze the patterns that such models generate. Analyzing the rules used to define linkages between individuals from the output of complex mechanistic models and seeing how these relate to processes involved in the population dissemination of infection could be especially helpful in designing questionnaires that capture the most useful definitions of linkages. The questionnaires may ask respondents whether specific conditions meeting such definitions exist. These may involve time spent at places and behaviors engaged in. Of course having information on the details of contact processes might be better. But, as discussed in Section 5.4, it is exceedingly difficult to get detailed data to describe networks. Thus simplification of questionnaires in ways that still capture the essential elements needed for robust inferences from network model analysis is important. Output from complex dynamic network simulations has been used to generate fixed networks from the dynamic network output for sexually transmitted infections (76). In this work, fixed network linkages that are directional are defined by timing of dynamic linkages that are non-directional. This approach could be used for questionnaire simplification. When transmission is not direct but via surfaces, water, food, or any means that can take an infectious agent across time and space from one individual to another, the need to model the media itself may arise. For example, when water carries infectious agents between individuals, it has been judged important to incorporate the water into the transmission models (77-81). Network graphs then become bipartite with humanmedia arcs but not human-human or media-media. Models that incorporate mechanistic details about how potentially infection transmitting linkages between individuals get generated and broken, or models that specify the paths that infectious agents take through different media to connect individuals may also be used directly in a robustness assessment strategy. They of course relax the simplifying assumptions of models without these mechanistic details.

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2.8. Statistical Analysis of Network Structure Two types of statistical analysis are relevant to inference robustness assessment. The first involves statistical analysis of network conformation parameters from data on who is connected to whom. We consider two ways such analyses can be used in inference robustness assessments. The second involves statistical estimation of infection transmission system model parameters by fitting network conformations to observed patterns of infection in a population. Both types of statistical inference might be integral to causal or control action inferences. When one is making causal or control action inferences about particular situations, one must consider a range of possible network conformations that might represent the particular situation one is modeling. Statistical analysis of network data on who is connected to whom might assist with this task. This type of data is obtained in various ways as outlined later in the Section 5.4. All of the different methodologies for gathering this data are problematic. But any available information of this type deserves to be integrated into inference robustness assessments. One way to use the output of analyses of contact patterns from contact histories is to enter the full range of network parameters consistent with observations into an inference robustness assessment. For example, the effect of age contact structures on inferences about what age groups are the principle transmitters of respiratory infections has been evaluated using this approach (82). As another example, consider inferences about whether a contact tracing strategy will be effective. Network model analysis might show that this depends on how much clustering of contacts there is in a population. The degree of clustering might then be assessed statistically from contact histories. Another way to perform inference robustness assessments using statistical models of networks is to use statistical models that make different assumptions about the underlying shape of the data. For example, clustering might be assessed using different statistical models that make different assumptions about the forces leading to clustering. Then the estimates from these different models might be used in a robustness assessment.

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Measures of transitivity and connectivity are especially relevant to epidemiology. One productive approach to statistical estimation of relevant transitivity and connectivity parameters builds upon exponential random graph models (83). Generalizable descriptive statistics of network characteristics like connectivity may be useful for inference robustness assessment in a couple ways. First, they may generate hypotheses that lead to inferences whose robustness deserves assessment. Second, such descriptions might help understand how contact processes lead to effects on infection transmission systems through effects on network conformation. These do not fall into the main inference robustness assessment activity of relaxing simplifying assumptions. But they help rationalize when and how to undertake that activity. With regard to estimating transmission system parameters by fitting network models, MCMC Markov Chain Monte Carlo (MCMC) methods have demonstrated both theoretical and practical utility (84-89). A particularly promising development is the integration of network structure modification steps directly into MCMC estimation algorithms (84). The approach taken is to impute missing information in the form of a graph that describes potential infectious contacts between individuals. The graph may initially be random but then connections can be switched in iterative steps one at a time until some stability in posterior distributions is achieved. This was done for estimating transmission probabilities within and between groups, for estimating transmission thresholds, and estimating the number of network connections within and between groups. These later entities represent the imputed missing data derived from the posterior distributions of an MCMC process that adds and subtracts network connections in iterative steps. While it might seem that such an imputation would be quite demanding, the limited range of possible networks that are consistent with the data gives this approach practical utility. MCMC output is in a particularly useful form for inference robustness assessments. When comparing two control action choices, the full range of posterior distributions consistent with the data can be sampled and the frequency with which one choice or the other is best can be observed.

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A different sort of interaction between inference robustness assessment and statistical analysis is the use of model transitions to simplify the statistical models of transmission systems. Models with too much realistic detail may not be appropriate for statistical analyses either because they have too many parameters that need estimation or because the extent of realistic detail in the model makes calculations too burdensome for practical purposes. Inference robustness assessment might first determine what type of detail in a model is needed for robust inferences. That might entail a model that is too complex and detailed for practical use in a parameter estimation procedure. Then a variety of model reduction approaches could be used to generate statistical models that are more practical for parameter estimation. Such reduction approaches might impose diverse simplifying assumptions that are known to be incorrect. For example complex models with detailed movement parameters for different classes of individuals might be reduced to either mass action or network models. If the use of estimation models making different model assumptions makes no difference for an inference, robustness has been demonstrated. 2.9. Designing Network Models with Robustness Assessment in Mind What can network scientists working on transmission system analyses do to facilitate integration of their models into an inference robustness assessment framework? First, they should clearly separate macro-network structure issues from micro-network structure issues. Micro-network structures that are generated by contact processes should be the focus of network analysis. The social processes and spatial relationships determining macronetwork structures are most likely unique in each situation one might model. Thus one will want to make hypotheses about and explore the consequences of differences in macro-network structure. As just discussed, that is currently easier in a deterministic compartmental model context. Therefore network models that facilitate transition to compartmental models that capture macro-network structure is preferred. Compartmental model macro-network structure is generated by distinct

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rates of contact between different risk groups. Therefore the network model should generate its macro-network structure in the same way. Second, network modelers should identify and state clearly all model assumptions. The failure to recognize the effects of model assumptions is the major cause of faulty inferences. The failure to recognize model assumptions is the main reason for failing to see their effects. Once model assumptions are made clear and the consequences of their realistic violation are considered, then inference robustness assessments follow naturally. 3. Nucleotide Sequence Traces through Contact Networks Nucleotide sequencing is cheap, fast, widely available, and becoming more so. How to best use it is yet to be defined. Epidemiologists currently use it to make inferences about whether transmission could have occurred between two individuals and/or determine which set of individuals were infected from a common source (90). That provides useful information about transmission systems but for the same reasons we will discuss in the Section 5.4 regarding collection of histories to analyze networks of contact, we will never be able to get an overall view of the shape of the network from such individual level deductions. We need population inference processes. Molecular information about infectious agents isolated from individuals can be used in two ways to assist population inferences about networks. It can be used to classify individuals or to establish genetic distances. Individuals can be classified as having agents that are mainly circulating in one population or another. This has been helpful for Tuberculosis (91,92), sexually transmitted infections (93,94) and HIV. HIV strains characteristic of intravenous drug users, homosexuals, and/or heterosexuals can often be identified. But the strains characteristic of one group can be found in some members of other groups. Such distributions of genetic types by risk group can be fit by models to estimate model parameters. Restriction Fragment Length Polymorphism (RFLP) analysis of microbial genomes of Tb organisms from individuals of different ages was used to classify organisms into small groups of identical strains.

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Clusters of identical Tb strains were analyzed with regard to age and gender structure in relationship to the overall population of infected individuals to infer what age groups and genders are infecting what other age groups and gender with tuberculosis (91,92). For STIs, strains of Chlamydia were classified from individuals that were linked by extensive but quite incomplete contact tracing (93). This helped fill in some gaps where linkage information was incomplete and to check for the consistency and inconsistency of histories not just in individual linkages but with regard to the pattern of linkages. Similar work has been undertaken for gonorrhea (94). Genetic distance data indicates how many transmissions there might be along the transmission tree that led from one individual down two separate chains of infection to two individuals whose distance is being measured. Genetic distances can be used to analyze the shape of genetic trees and therefore of transmission trees. The transmission trees generated by models can then be fit to the patterns of genetic trees for the agents. A task network scientists should undertake is to extract the information on transmission trees in a way that helps inferences about the conformation and behavior of infection transmission systems. It seems there should be information on both micro- and macro-network structure in nucleotide sequences.

A

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Figure 1. Consider the traces that could have been provided by genetic analysis of four infectious agents from individuals A through D. Infection has percolated to these individuals through network nodes consisting of other individuals in whom the agent first replicated and then transmitted. Genetic analysis will reveal that agents from C and D are closely related to each other and equidistant from the agent of B and only distantly related to the agent of A.

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The genetic tree corresponding to the transmission tree can be inferred using phylogenetic analysis methods. The tree will be inferred with error. But the extent of random error can be quantified such that the range of shapes to that tree can be specified. Thus a record is kept in the sequences of infectious agents regarding what elements their paths have in common and where they diverge. We will point out in Section 5.4 how difficult it is to piece together chains of contact from questionnaire. The infectious organisms, however, have been going around keeping a record of where they have been. Just as the human genome has demonstrated the migratory patterns of humans out of Africa better than any history, pathogen genomes are structured by their transmission histories and they can document that better than any set of contact histories can. Network analysis should take on the challenge of extracting that information in ways that allows inferences about the contact network shapes that could have generated the transmission tree shapes. There are a number of epidemiological questions that might be pursued from analysis of nucleotide sequence data. These include: Which groups are involved in sustaining transmission? Which are only being peripherally infected? Which are amplifying and disseminating transmission? There are similar questions relevant to risk factors: In what categories of individuals will changes in risk factors generate the broadest public benefit in the total population? Risk factor changes might involve behavior changes, sanitation intervention, hygiene improvement, vaccinations or infection treatment. Conversely we can ask in what categories of individuals will interventions have little value because they will become infected by some means despite the intervention and still carry on the chain of transmission. One key to extracting such information is linking sequence data to macro-network data describing contact rates between risk groups. To date, nucleotide sequence data has been available mainly from convenience samples not linked to other network data. Thus the assumption of random mixing has been employed and inferences sought have been restricted to past patterns of infected population sizes (95). If network scientists find ways to extract the information on networks from their nucleotide sequence traces, then epidemiologists will design studies

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to collect the data and insist that bioinformatics data bases are linked to network data. 4. Risk Factors for Transmissibility Controlling the spread of infection is a goal of public health. Epidemiology is the basic science of public health relevant to this task. But epidemiology in the last half of the 20th century focused on individual risk factors and, at least in the United States, defined itself by the use of methodologies that assume there was no interaction between individuals and thus assume away system behavior driven by networks of interaction between individuals. The dominant methods of epidemiological analyses assess risk factor effects using models that assume the outcome of one individual is independent of outcomes in other individuals. That is to say, they assume away infection transmission along with any network connections between individuals. This methodology was driven by chronic, non-infectious diseases. But it is also applied to infectious diseases and their control. It has had the nefarious effect of causing epidemiologists to ignore the most important risk factors – those that affect contagiousness and those that affect network conformation. Epidemiologists seek causes by comparing the experience of individuals with and without a risk factor. This can only detect risk factors that increase an individual’s susceptibility. Study designs that compare the infection experience of individuals exposed to source case infections with different risk factors affecting contagiousness are rare and expensive. Infection transmission system modelers seem to have almost the opposite problem. The main scientists modeling infection transmission as a causal system phenomenon at the population level have not been trained as epidemiologists. They are mathematicians and mathematical biologists. But these scientists have largely ignored the importance of risk factor identification and elimination in the control of infection transmission. Most of the work by mathematical biologists on analyzing infection transmission systems in the 20th century used methodology that made it hard to analyze data to detect contagiousness or transmission enhancing environmental risk factors. They did not model individuals on

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whom risk factor data could be assessed in a manner familiar to epidemiologists. The standard approaches to infection transmission system modeling are illustrated in a couple of very helpful texts (41,49). Noteworthy, however, these texts do not deal with the analysis of transmissibility related risk factors. Using discreet individual transmission models for epidemiological data analysis could potentially provide methods to determine the roles of risk factors in transmission systems, including risk factors that increase contagiousness. That is a highly worthy goal. Let us consider why. Part of the importance of risk factors that increase contagiousness is that there is more variability in contagiousness and more potential for control of risk factors that generate such variability. The greater variability of contagiousness derives from both behavioral and biological factors. The greater potential for control involves more modifiable behaviors, the potential of treatment to decrease contagiousness but not susceptibility, the potential for decontamination efforts to stop transmission, and correctable hygiene and sanitation deficiencies that affect contagiousness. Further increasing the importance of risk factors that increase transmissibility is the fact that they have greater system effects than risk factors that increase susceptibility by the same amount (96). That is because the most susceptible individuals are the first to get infected and the consumption of these susceptible individuals then slows transmission. There is no comparable consumption of the most contagious individuals. Identifying risk factors that affect transmissibility is tightly related to identifying modes of transmission. Classification of transmission modes can be made in diverse ways. Transmission modes might include the following and many more categories: •

• •

Airborne transmission where infectious organisms stay suspended in air through the formation of droplet nuclei that can travel considerable distances Droplet transmission where organisms settle quickly out of the air and contact must usually be within 1.5 meters to be effective Fecal-oral transmission involving any route, direct or indirect, from feces to mouth

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• • • • • •

Skin to skin transmission through direct contact Sexual transmission Surface or fomite mediated indirect transmission from either droplet or skin sources Water borne transmission Food borne transmission Vector borne transmission involving intermediate, often arthropod, hosts.

Most infectious agents are transmitted via multiple modes. Epidemiology has developed many methods to determine which modes of transmission exist but none to determine what roles the different modes of transmission play in the transmission system. Although we have long known that various common cold viruses are transmitted directly via skin to skin contact, and via air and surfaces, we have little intuition as to what effects interrupting just one mode of transmission or the other would have on population levels of infection. Likewise, although we know that HIV can be transmitted via either oral and anal sex, the population effects of reducing these modes of transmission is highly controversial with few helpful analyses. The traditional epidemiology approach allows for specifying modes of transmission by determining how contacts of different kinds are associated with infection risks. For example the association of anal sex and intravenous drug injection with HIV infection has been observed with enough consistency and control of confounding factors to convince epidemiologists that HIV can be transmitted via each of these modes. But because the association methodologies used to ignore network structures, the role of different modes of transmission in a transmission system cannot be determined by those methods. Network structure can cause the reduction of contacts via one mode to have markedly different effects from the reduction of the same number of contacts via another mode. Consider an example where one group is sustaining transmission via mode A of transmission and disseminating it to another group via that mode that transmits mainly via mode B. If chains of transmission via mode B die out quickly, then eliminating a specified number of transmissions via Mode A could cut off extended

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chains of transmission while eliminating the same number of transmission via mode B would have much smaller indirect effects. This may be the case for waterborne vs. direct transmission of many enteric pathogens, (77-79,81) anal vs. oral sex as well as transmission during early vs. late infection for HIV (97,98), and possibly airborne vs. direct transmission of respiratory infection. Traditional epidemiology cannot address these issues. A systems analysis approach is needed for this. Network models might have multiple links between individuals corresponding to different modes of transmission in individual network models. In bipartite graph models where sites connect individuals and vice versa, different sites might have transmission weighted by the modes of transmission involved. The first approach might facilitate analysis of individual risk factor effects while the second might facilitate analysis of environmental risk factors. 5. Overview of Infection Transmission Network Models Let us now consider the elements of network models in light of the three priorities for network analysis of infectious diseases. We first consider infection processes within hosts and then transmission processes between hosts before considering the standard issues covered by most Chapters in this book including choice of interacting partners in a network model, the choice of interaction events, methodological issues, computational issues, network macrotopology, network microtopology, spatial issues, dynamics, and control. 5.1. Infection Processes and Infectious Agent Characteristics A major division of infection processes that affects model structure is whether infections are microparasitic or macroparasitic. In microparasitic infections the infectious agent replicates in the host such that infectious load and transmissibility of an infected person is assumed to be independent of the total exposure dose and exposure doses are not cumulative over time. Microparasitic infections also have the characteristic that re-exposure to more infectious agent after infection begins is irrelevant because such exposures are always far less than the

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infectious load an individual already has. Most viral and bacterial infections and many unicellular parasitic infections are microparasitic. In macroparasitic infections the infectious agents acquired from the environment do not replicate in the host but rather each acquired agent constitutes a new source of infectious agents that can be transmitted from one host to another or indeed back to the original host, usually via some environmental contamination or vector. In most macroparasitic infections acquired immunity does not significantly inhibit the acquisition of additional infectious agents by the already infected host. Most infections with helminthes such as intestinal worms are macroparasitic. Most network models are of microparasitic infections. Within these, the temporal patterns of infection and of acquired immunity dictate different model assumptions. The simplest standard model assumes that upon transmission an individual goes from a uniformly susceptible state (S) to a uniform infected and infectious state (I) and thereafter to a completely immune state (R). This SIR model form is useful when seeking initial insights into system behavior for some infections. Its realistic violation, however, can lead to marked changes in system behavior. Almost always there is an incubation period (E) during which an exposed individual has acquired an infectious agent but has not yet become contagious. Thus an initial realistic relaxation of assumptions is to transit to an SEIR model form. More importantly, no infections induce complete and everlasting immunity. Thus elaboration of the R state is often called for in robustness assessment. Some infections, such as measles, have seemed to induce nearly lifetime immunity but elimination of infection in the population has revealed that enduring immunity requires continual re-exposure to the agent to boost immune responses. All infections induce some acquired immunity. For a few, however, the immunity seems too negligible to be included in a model. Thus a simple SIS model form might be used in seeking insights about system behavior. Often the lack of immunity is due to the fact that infectious agents are so variable that immunity stimulated against one particular strain is not helpful in protecting against the vast majority of agents that are not distinguished by the sophisticated methods needed to detect the multitude of immune stimulation and immune response variations that are possible. This is especially true for infectious agents like gonorrhea.

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In this and many other common bacterial infections the evolution of different antigenic variations that affect immunity is so rapid and involves such intricate complexity that it makes little sense to distinguish different immune types. In many viral infections such as polio, and some bacterial infections such as those caused by Streptococcus pneumonia, the immune variants fall into a number of fairly cohesive categories with little cross-reactive immunity between them so that it becomes worthwhile to keep track of each individual variant in an infection transmission system model. The model assumption that a single unvarying infectious agent is involved is almost never true. It is the variations in the infectious agents that make the genomic sequencing discussed earlier a potentially useful approach to working out infection transmission networks. But in a few unusual cases the variations do not affect immunity very much. One of those cases is hepatitis A. Worldwide there is no meaningful difference in the immunity stimulated by highly variable hepatitis A viruses. Thus hepatitis A viruses that have only 60% homology in terms of nucleotide sequences can stimulate immunity to each other almost as well as each agent can stimulate immunity to itself. More commonly there is some cross reactive immunity to different variants of the infectious agent but this cross reactive immunity is not complete. It is difficult in most model forms to handle this situation completely. Model complexity explodes as different cross reacting strains are added to the model. One approach to handling these realistic effects is to capture the effects of continuing agent changes by modeling waning immunity across multiple immune levels (99). Compartmental models cannot handle agent and immune process diversity well. Even given modest agent and risk behavior diversity, the number of compartments can reach astronomical levels. Discrete individual simulations, including network model simulations and individual event history mass action simulations, are needed. As agent and immune response diversity increases, it seems likely that micronetwork structures would gain in importance. This speculation has not been examined to my knowledge. The increase importance would derive from the fact that as each variant moves through a population, it might do so on very sparse networks of immunity holes left by other variants. This

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could be especially important for bacterial infections and modeling of such complex relationships may be necessary to devise an effective strategy for controlling agents like those that cause otitis and sinusitis (99). 5.2. Modes of Transmission The ways that infectious agents get out of a host to begin their transit to another host are highly diverse and highly dependent upon both biological and behavioral characteristics of the host. Even within respiratory infections there is great diversity. Different mucous membranes may be affected. Skin can provide an exit route directly or by touching mucous membranes. Agents may be aerosolized at lower, middle, or upper levels of the respiratory tract with consequences on the size and survivability of resulting aerosols. The routes agents can travel to another host are also diverse. The effectiveness of different routes is not only a function of the agent, but also of environmental factors like humidity, sunlight levels, characteristics of the inanimate objects that might carry an infectious agent, and above all behaviors of those involved in transmission. Again just considering the respiratory agents, there will be diverse mixtures of the role of skin, surface, droplet, or aerosol mediated transmission. The extent of mixture may vary not only by agent but by environment. The routes an agent may take to enter a host and the effects of host factors on which routes might work are again highly diverse. Likewise infectious agents are highly diverse with regard to how well they survive in the environment as they transit from one person to another, how much they can multiply outside of the human host, and the number of organisms in an exposure dose that is required before transmission becomes likely. Given this complexity, modes of transmission, as discussed earlier, can be conceptualized in many different ways. Issues of how modes of transmission affect transmission dynamics have been glossed over in both epidemiological investigations and models of transmission. Differences between airborne and droplet spread or in the survivability of an agent in the environment are usually ignored. I believe these are important issues to address. When a new infectious

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disease like SARS emerges, the actions that will work to control it will depend on the mixture of modes of transmission an agent has and the mixture of environments where different modes can act. To focus control efforts, the mode of transmission must be known. When deciding on what actions designed to reduce contact to control transmission, the relative importance of transmission in crowd settings vs. intimate gatherings will vary markedly by mode of transmission. From a modeling standpoint, a first decision in addressing modes of transmission is whether to incorporate the media into the model or not. The models of the Virginia Tech group originally incorporated media. Individuals go to a site and contaminate media or pick up contamination from media at the site. In those models, great complexity was added to the parts of the model that specify to which sites individuals go. Little specificity is added to media contamination to correspond to specific differences in airborne, droplet, skin, or surface spread. This seems like a serious imbalance that needs to be addressed by proper robustness assessments. It seems likely that important inferences might not be robust to tacit model assumptions made about the mixture of transmission modes in models that do not specify these. These deserve field investigation so that modelers can specify them in their models. But even without field data, inferences might be assessed as to their robustness to different spacetime patterns of contact between people that are involved in infection transmission. The appropriateness of decisions on what details to include or leave out will depend on the inferences that one pursues. If one were focusing on how to extrapolate information from transmission in a hospital to transmission at community sites, one would most likely need more details with regard to the media. If one were focusing on how resources should be directed to populations with different geographic and social structures, perhaps the movement of individuals would be more important to insure inference robustness. A second modeling decision depending upon modes of transmission is whether to make contact symmetric or asymmetric. There are two aspects to symmetry. The first is symmetry as to who is contagious and who is susceptible. The second is how timing of contact or timing of

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movement of individuals affects who can transmit to whom. The first type of asymmetry is generated both by biological and behavioral factors. For organisms transmitted fecal-orally, there are a few individuals who will contaminate media like swimming water with feces and many individuals who will take up the media orally. Thus asymmetric transmission should be considered. The same may be true for airborne transmission as only individuals with respiratory tract conformations that will aerosolize infectious agent might put many agents into the air while anyone breathing can pick the agent out of the air. This first type of asymmetry can be handled by specifying contagiousness and susceptibility differently for individuals with different characteristics. This can be done within the context of models that do or do not incorporate media and that employ structured mixing or metapopulation movement mechanisms for determining population patterns of contact. The second type of asymmetry arises because one person comes in contact with media before another person. The latter person cannot transmit to the first but the first can transmit to the latter. A judgment needs to be made as to whether such asymmetries will balance out in a manner that does not require their modeling. For airborne transmission, that seems likely. As the survival of the organism increases and the dilution rate of the media decreases, asymmetries may become more important. Also if the social structure of contacts generates a particular order of contact, such as might happen in needle sharing, then directional asymmetry will be more important. Since enteric organisms like rotavirus and norovirus survive a long time in the environment, robustness of any inference to symmetry assumptions would deserve greater consideration than for organisms like Shigella sonnei that will die out quickly. Models that incorporate media and use metapopulation movement formulations have a built in mechanism for generating directional asymmetries in contact. Models without media or that use the statistical mechanics assumptions of structured mixing cannot formulate directional timing of contact. Network models that begin with undirected arcs and that can be readily modified to add patterns of asymmetry that might be consistent

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with different hypothesized modes of transmission might be the easiest way to assess whether temporal asymmetries in contact processes could be important to consider in inference robustness assessments. Incorporating media and movement to handle asymmetries is a much more complicated task. 5.3. Interacting Partners in Infection Transmission Network Models It has already been explained how the choice of whether or not media is to be modeled determines whether or not the interacting partners will be pairs of individuals or individuals and transmission media. In order to fit into network theories and percolation analysis algorithms, network models with discrete individuals as the interacting nodes and undirected arcs are often chosen. In the network model discussed earlier of Mycoplasma pneumoniae transmission in a psychiatric hospital, bipartite graphs of contact sites (wards) and individuals were constructed but rather extreme assumptions about the mode of transmission were made by not specifying any mode of transmission (4). In order to enhance analytical power, the assumption was made that transmission to a site to an individual was independent of how many infected individuals there were at a site. That might apply to agents that are aerosolized at a very high level and that require very low doses to cause infection. It does not correspond to droplet transmission which is one of the principal modes among several modes via which Mycoplasma pneumoniae can be transmitted. Making assumptions that enhance analytic power is always justified when the goal is merely to gain insights into system dynamics or behavior. But even general insights can be wrong or misused in pursuing modeling goals. Thus it is worthwhile considering what could lead to lack of robustness for a percolation analysis or other network analysis. When infection transmission is via skin to skin contact, via air, via droplets or via contamination of inanimate objects that can carry infection from one person to another, then undirected network models may be appropriate. Even then, if the organism survives for any time outside of the hosts modeled as nodes, then inferences based on network analysis of undirected arcs between hosts might lack robustness.

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Likewise, if behaviors involving contamination or picking up of contamination are socially scripted such that one class of person performs a contaminating act and another performs an act that picks up contamination, then inferences from analyses employing undirected arcs may not be robust. Whether direct transmission or transmission via a media is assumed, the connections between individual hosts are defined by the existence of an ongoing potential for the mode of transmission involved to carry infection from one person to the other. Abstracting such connections into permanent connections between hosts is an extreme simplification that is never closely approximated in the real world. It is this abstraction, however, that brings the tools of network analysis to bear on infection transmission. The appropriate definition for a fixed arc that relates such a model to the real world is problematic but we leave that for discussion in the “methodology” Section that follows. 5.4. Methodology The book “Network Epidemiology” (100) puts field studies of network patterns into three levels. These all seek to describe micro-network conformations using data obtained in interviews about contacts. 5.4.1. Micro-network Interview Data The first level of study seeks merely to describe local networks around randomly selected subjects by asking them about their contacts. An example where graph theoretic considerations have helped derive useful population measures from such data deals with concurrency of sexual partnerships (101). Given consistency with mixing assumptions, it was shown that inferences of epidemiological importance about population network patterns can be made from individual level data (101). Specifically the extent that concurrent links are linked together can be inferred. The second level of study seeks to describe partial networks by following out the contacts of specific classes of individuals. The typical methodology used is to have subjects name their contacts and then go to

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these contacts and follow out their contacts. The only infections for which this is attempted to any extent that might be useful for constructing a network are sexually transmitted infections (STIs). Various strategies to trace out distant links and try to recompose them into a picture of the entire population have been developed for STIs. Tracing chains uses a strategy that keeps the sample size approximately constant as one goes various generations out. Snowball sampling follows out all contacts so the sample snowballs in size. Only a handful of studies have pursued either approach and none have achieved very complete descriptions of contact networks using this approach. It is rare that more than half of contacts can be traced in such an effort. For the very high risk populations where construction of the network is most important, 10% tracing represents a good effort. Thus there are many chances for biases and distortions to arise in networks of direct contact constructed from contact tracing data. An outstanding study of this type examined networks of contacts in Colorado Springs, Colorado that could have spread HIV infection (102). Eventually 3% of the Colorado Springs population had entries in this network. Many useful observations about the characteristics of the network were made, including the observation of a surprisingly close relationship between social distance and geographic distance in the key populations that could spread HIV infection (103). Another useful study of this type was conducted in Manitoba (93). The third study level accomplishes complete network descriptions by gathering data on every individual in a population in a manner that allows specification of other individuals to whom each individual is connected. Only a handful of such studies in general populations of any size have been conducted. One of these was conducted in the Nang Rong area of Northeast Thailand. The data in Nang Rong accrued in various steps without being designed at the start specifically for that purpose. The data permits some description of how the network evolved over time (104). This network data was used in a general heuristic manner by the Emory group within the MIDAS program to assess the chances for controlling an emergent H5N1 epidemic where this agent breaks out of its numerous avian hosts in Thailand to begin human to human transmission (74). The idea is that it might be useful to stock anti-

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influenza drugs and use them intensely so such an epidemic would not start a pandemic that might bring high levels of mortality and social disruption. One study in a subpopulation of school age youth was able to describe networks at the school level with relative completeness (105). This was the adolescent health study directed by the Carolina Population Center of the University of North Carolina. Since the dominant social ties of middle and high school students are to other students, this provides a nearly complete description of ties within one age group but not outside it. The study enquired about social and romantic partners in a questionnaire administered to everyone on one day and then pursued more details in subsequent studies of selected subjects. Useful insights relevant to adolescent health and the structure of networks have been made from this study data but there have been no formal transmission model analyses performed. The challenge for network scientists is how to use limited and potentially biased data of this sort to make robust inferences. As statistical methodology improves for integrating such data into infection transmission system analyses, we can expect epidemiologists to pursue the collection of more such data. The trick will be to get data and methods development into a positive feedback loop where improving methodology encourages more support for data collection which in turn will justify more methodological development. Emphasis in methodological development for using the easiest to collect data, namely egocentric data, will do the most to stir new data collection. Analysis of such data within an inference robustness framework that assesses the effects of assumptions about further details in micro-network structure and assumptions about macro-network structure might then justify the collection of level two types of data for the micro-network structure issues and contact matrix data for the macro-network structure issues. A fundamental problem with interview data methodologies is the definition of contact that is relevant to infection transmission. For sexually transmitted infections interview data may be attainable using acceptable definitions of what constitutes contact and people may be able to remember contact events with some reasonable accuracy. But for other infections this is highly questionable. Currently, for example, the

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MIDAS projects are building influenza transmission system models without addressing the issue of how different types of contact and different environmental conditions differentially affect airborne transmission which may occur over considerable time and distance from droplet spread infection which will have more restricted time and space dimensions. Some contact pattern studies have used data on who has spoken to whom under the assumption that if individuals have spoken, they could have transmitted infection (82). But given that most respiratory infections are spread by both airborne and droplet spread mechanisms and the nature of contact for these two modes is quite different, it is not clear how good this definition might be. No matter how far off transmission networks are from conversation networks, the use of conversation networks is superior to the use of assumptions about contact patterns that have little empirical support. In this regard, interview obtained contact pattern assessments using data on who has spoken to whom were found superior in transmission system analyses for making inferences regarding the effects of age specific contact patterns on the dissemination of droplet spread infections like mumps (82). The methodology used assessed the improvement in prediction of infection patterns from using the interview based data compared simpler assumptions about mixing. Clearly this fits within the inference robustness assessment methodology we advocated earlier. If we fail to develop methodologies that distinguish which modes of transmission are acting under different conditions and what role these play in a transmission system, we are less likely to control any new emerging infection. Ideally for an emerging infection, one would like to determine the modes of infection that are likely to be acting in the general community from data that can be gathered in the hospital setting. In order to do that, we will need definitions that distinguish contacts with differential risk for airborne or droplet spread transmission in the community. We will then need a combination of environmental and epidemiological data in the hospital setting to characterize the infectious agents as to their relative propensities to transmit via these modes. This can all be done. It is an issue of how much effort and resources are dedicated to the task. Perhaps this can all be done more efficiently by integrating nucleotide sequence data more fully into the task.

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5.4.2. Macro-network Interview Data In addition to these methodologies that focus on completing micronetwork structures, various approaches can be used that seek only to describe macro-network structures. One such approach is to have individuals identify characteristics of their partners that they can readily ascertain and that the investigator can use to classify the subject as well. This has been called the matrix construction approach (100). Another approach is to have subjects identify the sites they visit where contacts could be generated and then to make assumptions about the encounter and linkage processes that leads two people at a site to make a contact and then to engage in behaviors that might entail transmission. This has not been widely attempted but anecdotal experiences indicate that it might be useful. It is reasonable to think it might be subject to fewer biases than the partial network description approaches. It will be subject to the validity of assumptions about contact process at a site, but the robustness of inferences to various violations of assumptions can be readily assessed. 5.4.3. Environmental Contamination Data Data on the level of environmental contamination can help construct and analyze models either by specifying the potential for different modes of transmission to act under different conditions or as data the model can fit. Media sampled could be air or surfaces or vacuum cleaning filters. Environmental scientists and epidemiologists have only used such data for the first purpose. For that purpose it is important that organisms that are identified in the environment be viable organisms that could start an infection in a new host. Methodology for determining viability is expensive and consequently few studies have been conducted of environmental contamination levels except in special media like food and water where safety for consumption must be insured. On the other hand, it is cheaper to identify genome segments without regard to viability. If models have compartments for both viable and nonviable infectious agents, they can be fit to such data.

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5.4.4. Nucleotide Sequence Data Nucleotide sequence data for some infections is being extensively gathered for reasons not related for transmission analysis. Such data might be useful for model analyses. For example most HIV infected individuals get their reverse transcriptase and polymerase genes sequenced in order to assess the potential for resistance to emerge to each of the different antiviral agents that might be used. Such sequences are probably not as valuable for epidemiological analyses as envelope gene sequences. But they could still have great value. For the most part, however, sequence data relevant to transmission analysis is unlikely to be gathered until modelers demonstrate that making robust control decisions depends on it. The first step in that direction is to develop methods to use information on transmission tree patterns in transmission system analyses. A key issue for using nucleotide sequence information in transmission system analyses is to link the sequences to data on where the person experiencing the sequenced infection is located in the system. Such data linkage is unlikely to be available without special studies. The most likely data to be available will characterize individuals by their risk factors. If models classify individuals by these risk factors, then patterns of genetic distances between individuals with different risk factors can be compared with patterns of transmission distances generated by models. 5.5. Computational Modeling Network model simulations of infection transmission have been constructed in “agent based” simulation languages like Swarm (106), Ascape (107), and Repast (108). The Hopkins-Brookings Institute group in MIDAS uses Ascape. These are all higher level languages for general construction of agent based models available at modest cost or free. Epidemiologists, however, generally do not have the programming skills needed to use these languages. Commercial modeling packages like AnyLogic (109) have proven to be valuable for model construction by researchers who are not sophisticated programmers. Various other high level simulation programs have been developed and made publicly

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available for specific infections. Probably more than 20 different general programs for discrete individual simulation of infection transmission that were intended for public use and intended to be adaptable to particular infection problems have been developed. In general, however, each question is unique enough so that it is hard to use these general programs. To date, the most productive researchers have developed code that is narrowly applicable to the specific issues they are investigating. Good software might open up this field so that more epidemiologists get involved. But our effort to construct a general dynamic network simulation for sexually transmitted infections called GERMS (29) proved to be too complex to maintain and adapt readily to the multitude of unique questions that need to be addressed. We sought to simulate realistic processes of encounter and linkage formation and breakup for sexual relationships. Each time a relationship is formed, it changes the encounter and linkage environment of others. Therefore, we formulated all events as independent Markov processes and simulated one event at a time using mass action formulations in structured mixing settings to determine the sum of all encounter rates in a population. This process proved hard to tune so that specific network patterns were produced. An ideal model construction and analysis environment that this naïve epidemiologist would love to have is one that allows for compartmental model formulation of macro-network structure using structured mixing formulations. Analysis of such models using both numerical solutions of deterministic differential equations and stochastic simulations of discrete individual probability formulations should be possible. Then within the mixing sites the conversion of mass action formulations to network formulations with different micro-network structures should be facilitated. This is simpler than the dynamic network environment of GERMS (29) in that the mass action environment for encounter processes does not change with every event. The ability to transit between deterministic and stochastic mass action and network formulations would facilitate inference robustness assessment and allow one to make computational tradeoffs between memory and calculations in addressing specific problems.

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5.6. Macro Topology of the Network Most broad aspects of macro-network structure of any infection transmission system within either urban or rural areas remain unknown because of the difficulties pointed out in the methodology Section. There are a few advances, however, in this regard. Studies of the space-time patterns of common infections from surveillance data over many years have demonstrated very clearly that population contact patterns specified by urban size and degrees of connection between different urban regions are clear determinants of infection patterns (110-112). There has long been an interest regarding macro-network contact structures by age group and the patterns of childhood infections. Most contact matrices employed, however, have been arbitrary and unrealistic. Recently, however, interview data on contact patterns by age has been used in fitting infection transmission system models to observed patterns of mumps and of pandemic influenza (82). The conclusion was reached that the 13-19 age group plays an important role in spreading this sort of infection and should be a focus of control efforts for any emerging infection spread via droplets. The robustness of this inference was not fully assessed and there are reasons to think it may be wrong. This assessment of the effect of macro-network structure should clearly be reexamined using network models with different micro-network assumptions. But as this sort of assessment is done for an increasing number of infections, patterns that modelers should use in any robustness assessment for an inference will be increasingly narrowed. Transportation system data has been touted as a key guide for control of emerging infections (70,75). Analyses of transportation determined patterns of infection have assumed uniform susceptibility to infection. In an emerging infection that a population has never experienced before, this might be a reasonable assumption. But in most cases, immunity is a key issue to deal with in the context of population contact structure. One problem is how to separate out the influence of actual structure of contact rates between different population segments and the influence of immunity from previous infection having circulated through the same population structures and transmission systems in the past. Several investigators have tried to use travel data to predict patterns of spread for

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influenza through countries or the world (113-117). In these studies model parameters were fit to observed annual epidemic patterns of influenza. But cumulative immunity from one annual epidemic to another is clearly acting and this immunity has not been taken into account. In fact the spatial-temporal pattern of influenza emergence in any year are due to the speed in different areas with which the epidemic reaches the fairly high infection levels that have to be achieved before an influenza epidemic is perceived. That speed is a function of immunity levels. The pattern of spatial spread has little to do with the spatial temporal patterns of when epidemics are perceived. We have shown in unpublished analyses that this failure greatly distorts conclusions about the flow of infection and its control. One vision for eventually determining the macro-network structure of a population is to combine various sorts of information. These would include the type of data and modeling structure used by the Virginia Tech group, inferences about the macro-network structure made from the study of transmission of numerous different infectious agents and their nucleotide sequence patterns in the same population, sequential serological studies in selected individuals whose contacts within the transmission system are documented, and biological studies of the potential for transmission via various modes of the various agents studied. It is my belief that every health officer should some day be able to determine what institutions and social events bring people together to form the key network connections that could sustain and amplify circulation of various infectious agents in their population. It should be possible on the basis of such an analysis for a health officer to make decisions about changing contact structure, where to focus case detection and isolation efforts, where to focus chemoprophylaxis of infectious agents, where to concentrate tracing and quarantine efforts, and where to insure maximal vaccination coverage. When pandemic flu hits again, it seems quite reasonable that with such knowledge, even if control of transmission is lost, total infection rates and mortality could be reduced by as much as 80% if proper actions are taken. Currently we are a long way from that. But the future is coming fast.

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5.7. Micro Topology of the Network A correlation model analysis of childhood infections demonstrates that the micro-network topology of infection transmission systems brings infected individuals in contact more often with immune and other infected individuals than would be expected by chance (47). Analysis of the contact patterns generated by simulation based on transportation pattern influence on movement to mixing sites results in a similar conclusion (70). This significantly slows transmission dynamics, broadens out epidemic peaks, and reduces variance in infection levels (31). Expected patterns for sexual contact, airborne transmission, droplet spread, and transmission via inanimate objects or vectors should be different in this regard. In a purely heterosexual contact network there will be no triplets and only a modest increase in quadruplets where partners switch back and forth. In an airborne infection of respiratory infections at local mixing sites sharing the same airspace, all triplets involving people at the site will be closed and higher level structures like quadruplets or quintuplets will be saturated as well. On the other hand, as the population size in airborne contact in each unit goes up and the per person transmission rate goes down, the more the system will behave as if there is random mixing (31). But for droplet spread of respiratory infections, more direct contact will limit the formation of saturated small contact units except in households. There has been considerable work on following infectious agents through air, on surfaces, on hands, etc (118). But there is little field data and few tracer studies of particles in the environment that bear on issues related to how many people will be contaminated by someone in different environments. In summary, our knowledge of the microtopology of infection transmission systems is minimal. 5.8. Spatial Aspects Space is important to infection transmission at three different scales. On a global or large region scale, the space covered by air travel is important in the spread of emerging and other infections (116). Within localities,

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distance patterns to residences or contact sites play an important role (103). At transmission sites, spatial issues arise regarding the distance that an infectious organism leaving a host might travel to find another host (118). The distinction between the space-time dimensions of airborne, droplet, and surface transmission is an important issue that, unfortunately has not been adequately addressed. The lack of field information here may justify theoretical work on network effects that ignores this issue. But rather than ignoring this issue, it might help for theorists to incorporate it into inference robustness assessments in order to motivate appropriate data collection in this area. 5.9. Dynamics and Control Much of the theoretical work on infection transmission systems over the past 20 years has been demonstrating how contact patterns affect infection transmission dynamics. Comments relevant to this have been dispersed throughout this document and are especially found in the Section on micro-network topology. Likewise, comments on infection control are dispersed throughout this document. The Section on inference robustness assessment has this as a major focus. 6. Conclusion and Perspective In summary, theory for a science of infection transmission system analysis is flourishing, but data is languishing. One of the most exciting advances in the development of analytic tools for this science has been the incorporation of population structure into network analyses using percolation theory. But as long as such analyses must assume a fixed network, the robustness of inferences made will be suspect and will need to be assessed by realistically relaxing the assumption of fixed relationships. Addressing the data limitations may first require advancing analysis theory in ways that facilitate inference robustness assessments. When such assessments clarify what data are needed to make robust inferences, then the collection of such data is more likely to be supported. Particularly valuable data that relates more specifically to network

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models than to any other model forms includes nucleotide sequence data from infected individuals along with additional data on their position in the transmission system. Robustness assessments that consider how much more robust infection control inferences could be if they could be based on such data need to be performed to motivate the collection of such data. But first network scientists need to develop new methodology to incorporate such data into transmission system analyses. Recent work using pgfs in the construction and analysis of network models that permits analyses using minimal computations should give network model analysis a central role within the pantheon of model forms that might be used in an inference robustness assessment. The key to using network models in this way is to specify their assumptions and to specify the assumptions in other models that they can relax. A model does not have to specifically incorporate structures involved in transmission in order to relax assumptions about such structures. For example, different social relationships may imply different micro- and macro-network structures. Models do not need to incorporate these social processes to assess the validity of inferences with regard to realistic violation of assumptions about the effects of social structures on contact patterns. They just have to show that across a range of patterns that different social structures might be generating, the infection control inference or causal inference holds. The inferences assessed in inference robustness assessments need to be chosen carefully in order to advance support for this area of science. Network modelers need to work with epidemiologists to define control issues that they can address. As much as possible, modelers should couch the inferences they seek in terms of risk factor effects. That will facilitate use of model analysis based inferences by epidemiologists and better point the way to new data collection. Infection transmission system behavior and the effects of control measures are sensitive to a seemingly endless number of real world complexities. Every infectious agent is likely to circulate differently within the social structures that create potentially infection transmitting contacts between individuals. This is true even when the range of agents considered is limited to those causing respiratory infections because all respiratory infections are transmitted by a variety of different modes and

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INDEX

adaptive immunity, 343, 351 agent-based, 467 amplification, 33, 34, 37, 112, 119, 313 apoptosis, 351, 356 Arabidopsis thaliana, 90 average connectivity, 109, 110, 182, 183, 276, 395 B cell, 76, 342, 345–349, 353–355 Bacillus subtilis, 124, 273 bacteriophage lambda, 103, 104 Barabasi–Albert, 9, 153, 187 basal species, 392, 426, 427, 433, 435, 436, 439–442, 444 basic blocks, 37 Bayesian network, 44, 58, 60, 61, 63, 64, 69, 77, 102, 210, 246 Bayesian score, 71 best-fit, 14, 67, 208, 248 betweenness centrality, 2, 450 biocoenosis, 366, 370 bioenergetic model, 425– 427 bioinformatics, 96, 97, 164, 176, 216, 232, 476 biological networks, 2, 4, 8, 9, 16, 22, 23, 28, 44, 47, 52, 101, 114, 145, 148, 152, 213, 273, 274, 276, 280–282, 284 biomass, 14, 176, 182, 189–191, 235, 236, 366, 375, 376, 378, 383, 385, 386, 389, 394, 395, 400, 401, 427, 430 bipartite graph, 168, 169, 209, 466, 468, 479, 485 Boolean modeling, 99, 100 Boolean network, 42, 55, 56, 58, 67, 109, 208, 211, 213, 238, 278 Bose-Einstein condensation, 11, 12 building blocks, 23, 228, 325, 402 Caenorhabditis elegans, 3, 4, 139, 140, 150, 152, 263, 268 cancer, 29, 76, 93, 121 carnivore, 371, 426, 442 cascade, 87, 106, 117, 119, 133, 153, 233, 300, 358, 397, 401, 426, 430–432, 438, 439, 442 catalytic, 88, 98, 101, 168 causal dependency, 44 chemokine, 355

507

508

Index

chromatin immunoprecipitation, 93, 206 circuit feedback, 87, 110, 116, 117 integrated, 24 negative, 110, 115–118 positive, 110, 116, 117, 119 clustering coefficient, 3, 4, 8, 9, 11, 22, 177, 182, 186, 214, 272, 274, 276, 280, 322 co-evolution, 268 co-immunoprecipitation, 135, 137, 140 co-occurrence, 121, 138, 402 coexpression network, 202, 218, 241, 245, 247, 267 combinatorial, 23, 43, 67, 113, 259, 347 combinatorial explosion, 23, 36, 181 combinatorial transcription logic, 112 community structure, 120, 121, 375, 400, 405, 450 compartmentalization, 22, 101 complementary DNA, 89 complexity, 36, 37, 43, 48, 50, 52, 62, 64, 67–69, 77, 112, 147, 164, 174, 186, 199, 203, 232, 257–259, 266, 273, 283, 291, 293, 294, 311, 315, 322, 325, 327, 328, 350, 353, 368, 375, 380–382, 384, 386, 393, 423–425, 438, 443, 463, 467, 468, 481–483 component network, 200, 201, 203, 205, 207, 209, 210, 215, 217, 222, 224, 226, 233– 235, 240, 249, 250 composite network, 200, 203–205, 207, 209, 214, 221–223, 225, 226, 228, 232–234 computation, 44, 72, 181, 184, 294, 312, 314 computational, 36, 46, 50, 97, 99, 105, 112, 134, 138, 139, 152, 155, 199, 200, 203, 205, 206, 209, 213, 239, 244, 245, 248–250, 267, 291, 297, 304, 310, 313, 315, 323, 324, 327, 358, 361, 451, 479, 492 computational modeling, 97, 141, 175, 200, 208, 308, 491 connectance, 379, 382, 383, 388, 392, 393, 397, 426, 430, 433–436, 441 connectivity, 4, 11, 26, 33, 106, 108–110, 115, 144, 154, 165, 184, 186–189, 191–193, 224, 231, 272, 276, 277, 295, 296, 298, 311, 314–317, 319, 321, 322, 327, 396, 427, 430, 436, 437, 440, 471 contact group, 457 contagiousness, 465, 476, 477, 484 convergence, 50, 303, 313 conversion, 164–166, 177, 183, 184, 427, 492 correlation profile, 31–33 cortical networks, 315, 320, 322 cryptosporidia, 449 cytokines, 344–348, 350, 351, 353, 355, 356 cytoskeleton, 133, 356

Index

509

decoupled evolution, 268 degree distribution, 2, 3, 8–10, 16, 22, 32, 215, 221, 227, 318, 322, 395, 404, 440, 450, 463, 464 determinism, 104 deterministic models, 51, 54, 59, 105, 454, 455, 461 differential equations ordinary, 102, 103, 455 partial, 103, 104, 361 differentiation, 30, 116, 264, 345, 347, 353, 355, 356, 358–360 divergence, 153, 187, 263, 266, 268, 283, 305 diversity, 58, 104, 267, 282, 293, 298–300, 303, 304, 311, 327, 328, 341, 368, 380–382, 384, 388, 423–425, 438, 443, 444, 481, 482 DNA, 27, 28, 30, 45, 51, 58, 62, 63, 83–86, 89, 93–97, 104, 109, 111, 112, 114, 122–124, 133, 136–138, 141, 147, 148, 152, 164, 171, 172, 190, 220, 236, 249, 257, 261, 263– 266, 274, 282, 350 Drosophila melanogaster, 3, 139, 140, 150 drug targets, 153, 190 duplication, 10, 33, 35, 37, 153, 154, 259, 268, 270, 273, 275–277, 280, 283 dynamic models, 70, 427 dynamical models, 43, 50, 54, 63, 204, 234, 236, 369, 391, 400, 425 dynamics, 1, 26, 34, 36, 41, 50, 51, 54, 58, 83, 84, 92, 98, 106, 107, 109, 114–117, 120, 122, 134, 154, 155, 190, 191, 200, 201, 204, 208–211, 213, 214, 220, 226, 228, 229, 233–236, 238, 240, 241, 249, 250, 260, 273–275, 277–279, 282, 283, 292, 294, 296, 298, 299, 302, 303, 306, 307, 309, 310, 319, 320, 322, 325, 327, 328, 371, 374, 384– 386, 388, 399, 400, 402, 404, 405, 423–427, 429–431, 438–443, 449, 455, 456, 462, 463, 467, 479, 482, 485, 495, 496 ecological networks, 365, 366, 368, 369, 375, 380, 381, 383, 400, 402, 405, 406, 423, 425, 438, 439, 443 ecology, 2, 365, 370, 371, 373, 376, 378, 380, 382, 401, 405, 406, 424, 439, 444 ecosystems, 365, 370, 376, 378, 380, 397, 406, 423–425, 440, 443 edge, 32, 44, 61, 64, 72, 98, 107, 109, 115, 143, 184, 208, 210, 215, 217, 228–230, 276– 278, 280, 327, 390, 391 elementary mode, 178, 180, 181, 185, 189, 235 encoding, 47, 51, 54, 62, 64, 83, 88, 122, 123, 146, 240, 243, 247, 260 enzyme, 28, 86, 133, 163, 165, 167, 168, 172–176, 183, 188–190, 204, 209, 218, 235, 240, 241, 247 epidemiological, 458, 475–477, 482, 486, 489, 491 epidemiology, 471, 476, 478, 479, 486 Erdös–Rényi, 9, 22, 182 errors, 89, 176, 190, 435, 436, 441

510

Index

Escherichia coli, 6, 8, 13–16, 107, 109, 110, 118, 122–124, 139, 140, 183, 184, 186, 188, 189, 191, 192, 211, 226, 227, 235, 239, 241, 247, 248, 259, 263, 271, 282 eukaryote, 53 evolution, 21–23, 33, 34, 37, 104, 134, 138, 139, 153–155, 185, 187, 190, 193, 204, 214, 240, 257–262, 264, 265, 267–270, 272–275, 281–283, 365, 377, 405, 452, 481 evolvability, 23, 34, 37, 186, 263 extinction, 386, 396, 430, 458 extreme pathway, 179, 181, 189, 212 factor graph network, 63, 65, 66, 214 fan, 119, 230 feedforward loop, 65, 117, 118 fluctuations, 34, 67, 104, 323, 324, 387 flux, 13–16, 47, 164, 165, 177–179, 181, 192, 209, 211, 212, 235, 236, 238 flux balance analysis, 179, 180, 189, 191 fluxes, distribution of, 192 food web, 2, 3, 389, 395, 425, 426, 429, 431, 432, 434, 435, 437, 439, 441, 442 functional association, 22, 134, 138, 139, 202 fundamental pathway, 181 gene duplication, 34, 138, 153, 154, 187, 258, 262, 263, 268, 271, 274, 276–278, 283 gene expression, serial analysis of, 93 gene fusion, 138, 202, 226 gene neighborhood, conserved, 202, 226 gene transfer, horizontal, 138, 269 genetic distance, 473, 474, 491 genetic interaction, 86, 98, 107, 134, 137, 142, 144, 202, 203, 207, 220–222, 228, 232, 233, 239, 242–244, 249 genetic recombination, 187 genome sequence, 171, 172, 175, 176, 199 genomics, 2, 114, 154, 215, 216, 226, 251, 257, 260 genotype, 23, 212 geometry, 84, 122, 125 graph, 9, 27, 28, 32, 34, 46, 58, 64, 65, 71, 86, 88, 98, 102, 106, 107, 109, 110, 121, 168, 175, 177, 179, 182–184, 186–189, 192, 208–210, 213, 214, 216, 217, 223, 224, 249, 257, 269, 272, 274, 276, 317, 322, 367, 373, 382, 385, 389–391, 398, 461–464, 471, 486 directed, 102, 116, 118, 170, 184, 210, 269, 272, 317, 465, 466 undirected, 210, 367, 466 Helicobacter pylori, 13, 139, 140, 150 herbivore, 400, 402, 435

Index

511

heterogeneity, 21, 22, 122, 358 heterogeneous, 14, 21, 22, 28, 34, 106, 122, 125, 200, 201, 204, 205, 209, 210, 213–215, 223, 224, 226, 232–234, 238, 240, 243, 248, 249 hierarchical, 4, 5, 7, 8, 10, 16, 22, 26–28, 30, 35, 36, 76, 78, 186, 218, 271, 295, 326, 366, 403, 426, 438 hierarchy, 2, 4, 28, 29, 36, 117, 121, 347, 397, 423 high-throughput, 89, 93, 134, 136, 137, 139, 141, 204–207, 214, 239 homeostasis, 115, 116, 303, 304, 377 Homo sapiens, 139, 140, 150 homogeneity, 103 homogeneous, 14, 104, 107, 263, 315, 318, 321, 366, 459 hub, 29, 31, 32, 77, 145, 152, 272 hypergraph, 99 identifiability, 52, 62 immune network, 344, 347–350, 356, 357, 359 immune system, 341–348, 350, 351, 353, 355–359 infection, 344–347, 355, 356, 360, 449–452, 454–458, 461–463, 465, 467, 469–471, 474–476, 478–481, 483, 485–491, 493–497 infection control, 450–452, 465, 467, 496–498 inference, 42– 46, 49, 50, 52–55, 62, 64, 67, 71–74, 76–78, 100, 102, 105, 176, 293, 452, 454, 464, 470, 473, 484, 493 inference robustness, 452, 453, 460–466, 468, 470–473, 483, 485, 488, 489, 492, 496, 497 innate immunity, 342 integration, 10, 77, 201, 204, 207, 212, 222, 226, 237, 246, 265, 291, 293, 295, 296, 312, 313, 324, 328, 349, 356, 357, 424, 471, 472 interolog, 139, 150 invasion, 392 irreversible, 119, 153, 168, 170 isomorphic subgraph, 145 keystone predation, 381, 401 layered structure, 224, 225, 311 learnability, 49, 52 linear network, 56, 57 linear programming, 179–181, 192 link, 2, 9, 13, 16, 26, 27, 34, 58, 76, 97, 106, 114, 122, 144, 154, 165, 176, 202, 231–233, 243, 248, 277, 280, 293, 300, 315, 317, 322, 328, 348, 353, 369, 380, 381, 386, 392– 395, 402, 405, 426, 429, 468, 491 logical functions, 113, 209, 213

512

Index

logical modeling, generalized, 100 machine learning, 42–47, 49, 52, 66, 72–74, 77, 78, 280 macro-network, 458–460, 463, 472, 474, 475, 488, 490, 492–494, 497 macromolecular network, 114, 115 major transitions, 23 mass action, 384, 451–463, 467, 472, 481, 492 mass conservation, 166, 180, 192 mass spectrometry, 134–136, 140, 150, 172 mechanistic movement, 459 metabolic network, 3–6, 13, 16, 78, 144, 154, 164, 166, 169, 170, 174–179, 181, 184, 185, 187–193, 201, 206, 209–212, 218, 233, 235, 236, 240–242, 247, 248, 250, 274, 276 metabolic pathway, 42, 177, 179, 181, 183–185, 202, 204, 218, 226, 242, 247–249, 265, 271 metabolism, 13–16, 163–169, 171, 172, 174–176, 182–188, 190–193, 199, 203, 207, 209, 210, 212, 213, 218, 219, 233–236, 238, 239, 247, 248, 250 metabolite, 13–15, 168, 170, 174, 177, 182, 183, 185–190, 203, 204, 209, 233, 235, 238, 249, 274 external, 170, 179, 211, 213 internal, 170, 177, 178 metabolomics, 250 metapopulation, 459, 463, 467, 484 metazoan, 122, 134, 259 micro-network, 457–459, 462, 472, 481, 486, 488, 490, 492, 493, 495, 496 microarray, 15, 42, 50, 51, 89, 91, 92, 94, 95, 97, 134, 141, 172 microenvironment, 346, 350, 354 microorganism, 110, 191 microorganisms, 84, 111, 119, 190 migration, 343, 344, 349, 350, 355, 356, 366, 440, 444 modular architecture, 21, 22, 26, 37 modularity, 4, 10, 23, 25, 26, 31, 33–36, 84, 114, 115 module, 10, 22, 26, 30, 31, 36, 63, 65, 76, 111, 112, 114, 115, 120, 186, 225, 226, 232, 241, 242, 259, 265, 270, 271, 316, 353, 402 molecular complex detection, 121 morphological, 122, 267, 296, 303, 314, 399, 404 morphology, 295, 296, 298, 403 motif, 5, 6, 95, 97, 145, 146, 227–232, 242, 243, 266, 269, 271, 279, 312 multi-scale, 291 multicellular organism, 104, 111, 119, 137, 171, 173, 357 mutation, 137, 174, 187–190, 202, 266–268, 275, 276, 278, 280, 281, 349 mutualism, 367, 377, 379, 401

Index

513

network alignment, 148–150 network navigation, 450 network resilience, 450 neural network, 4, 57, 68, 291, 308, 309 niche, 378, 389–392, 397, 401, 403, 423, 425, 426, 430–443 node, 2–12, 22, 28, 33, 34, 61, 64, 106, 115, 116, 122, 143, 152, 153, 177, 182, 183, 217, 228–232, 241, 272, 276–280, 345, 346, 355, 360, 377, 436, 437, 440 noise, 22, 54, 55, 58, 59, 64, 67, 69–71, 77, 95, 104, 125, 147, 269, 320, 323, 324 nonlinear persistence, 438 normalization, 65, 92 oligonucleotide, 90, 91 omnivores, 426, 430, 435–437, 442 open reading frame, 84, 171 optimization, 13, 44, 45, 47, 66, 212, 219, 236 orthologous, 138, 150, 226, 242 orthology, 241, 242 oscillation, 118, 308, 352, 386 pattern formation, 152, 281, 361 peptide, 135, 136, 346 permutation test, 216, 217 petri net, 101, 168–170 phenotype, 23, 148, 152, 217, 244, 248, 267, 282 phylogenetic, 34, 138, 139, 171, 202, 226, 241, 245, 263, 265, 398, 475 phylogenetic co-occurrence, 202 phylogenetic profile, 138, 226, 241, 245 phylogeny, 392, 400 physical interaction, 137, 202, 220, 221, 233, 242, 243, 245 pleiotropy, 23 polymerase chain reaction, 92, 298 post-translational modification, 135, 172, 347 power law, 2, 11, 14–16, 22, 106, 108, 109, 182, 184, 186, 188, 193, 320, 327, 393, 395, 404, 464 predation, 367, 369, 381, 400–402 predator, 381, 384, 385, 391, 392, 401, 402, 423, 427–432, 434, 439, 443 prediction, 34, 43, 50, 63, 74, 75, 96, 118, 139, 204, 205, 220, 226, 236, 239, 240, 244– 248, 489 predictive, 75, 103, 125, 200, 201, 203, 205, 207, 211, 234, 240–242, 244, 245, 247, 250, 280, 393 prey, 136, 137, 367–369, 371–373, 376, 381, 384–386, 389–392, 397, 401, 402, 406, 424–428, 430, 437, 440, 442, 443

514

Index

probabilistic model, 58, 59, 75, 206, 324 probabilistic relational model, 63, 64 prokaryote, 53 proliferation, 30, 342, 345, 356, 357, 464 promoter, 146, 258, 261, 265–267, 271, 272, 350, 351 propagation, 32, 57, 69, 305, 320, 405 protein complex, 121, 137, 147, 149, 150, 206, 220, 231, 236, 248 protein interaction network, 3, 6, 15, 34, 78, 121, 134, 141, 144, 145, 147–150, 152, 153, 155, 202, 206, 220, 223, 224, 233, 236, 247, 269, 272, 276, 280 proteomics, 141, 172, 257, 260 random network, 4, 7, 9, 10, 110, 152, 185, 186, 214, 227, 277, 279, 317, 321, 322, 425, 438, 457 reductionist, 1, 291, 295 regulatory complex, 28 regulatory region, 45, 84–86, 95–97, 109, 111, 113, 260, 266–268, 271, 276, 278 relative persistence, 430, 431, 433 repertoire, 190, 298, 305, 306, 308 representation, 2, 7, 30, 43, 44, 47, 49, 54, 97, 115, 138, 142, 144, 156, 167–169, 207– 209, 232, 238, 239, 309, 367, 372, 390, 395, 398, 401 restriction fragment length polymorphism, 473 reverse engineering, 75, 205, 206 reverse transcription, 92, 298 reversible, 13, 153, 168, 170, 174, 183, 303 risk factor, 451, 468, 475–477, 479, 491, 497 RNA, 21, 41, 42, 62, 83, 85, 86, 88, 92, 113, 122, 133, 164, 172, 225, 259, 348 robustness, 21–23, 32, 37, 103, 152, 153, 174, 188–190, 211, 259, 278, 279, 395, 404, 438, 452–454, 456, 457, 459, 460, 462, 464–467, 469–472, 480, 483–485, 490, 493, 496, 497 Saccharomyces cerevisiae, 3, 6, 8, 14, 93, 95, 107, 109, 110, 121, 123, 139, 140, 144, 150, 152, 231, 239, 248, 268, 282 sample complexity, 52 sampling, 53, 54, 74, 394, 406, 487 scaffold graph, 31, 32 scale-free network, 7, 9, 10, 16, 76, 144, 152, 274, 464 self-similar, 186, 193 self-similarity, 186 sequence database, 135, 136, 147, 150 signal transduction, 114, 133, 149, 153, 233, 276 signaling, 106, 121, 133, 137, 141, 147, 156, 199, 202–204, 209, 233, 238, 248, 249 intercellular, 351

Index

515

intracellular, 347–349, 351, 356 simulation, 43, 50, 63, 115, 152, 233, 239, 273, 274, 296, 311, 430, 451, 491, 492, 495 small world, 22, 23, 28, 110, 144, 182, 184, 450, 464 social networks, 144 solenoidal, 123, 124 spatial, 94, 103, 104, 122, 123, 125, 225, 291, 295, 296, 312, 317, 320, 327, 359–361, 386, 406, 444, 459, 472, 479, 494–496 stability, 116, 269, 326, 370, 374, 377, 380–384, 386, 424, 425, 438, 471 state-space model, 59, 61, 71 static models, 43, 63, 397 stationarity, 54, 116 stationary, 51, 52, 67, 116, 122 statistical learning, 67–69 statistical validation, 45, 73, 74 steady-state dynamics, 235 stochastic, 34, 48, 51, 52, 58, 59, 62, 63, 67, 104, 125, 204, 209, 276, 277, 281, 309, 320, 324, 397, 426, 430, 431, 454, 455, 461, 468, 492 stochastic equations, 104 stochastic models, 58, 426, 431, 454, 455, 458, 463 stoichiometric, 13, 98, 101, 177, 181, 189, 210, 235, 271 stoichiometry matrix, 169, 170, 177, 179, 181, 186 Streptomyces coelicolor, 175 sub-graph, 3, 5, 6, 32, 231, 272, 274, 279, 280 subnetwork, 247, 258 substrate, 13, 86, 133, 165, 167–170, 173, 177, 183, 184, 186, 209, 239, 293, 304 synapse, 300, 302–304, 324 synaptic plasticity, 303, 304, 306 synthetic, 91, 137, 140, 184, 191, 202, 207, 220–222, 239, 242, 243, 245, 291, 293 systems biology, 41, 78, 126, 343 T cell, 342, 345–348, 351, 353, 355, 357, 358, 360 taxonomy, 116, 298 technological graphs, 37 temporal aggregation, 53 time series, 44, 50–53, 63, 67, 236, 239 tinkering, 23, 33, 37, 273 topological overlap, 25, 26, 28–30, 186 topology global, 106, 107, 111, 181 local, 107, 109–111 training, 43, 47, 50–53, 66–68, 70, 73, 75, 244 training sample, 47, 50, 52, 68

516

Index

transcription, 4, 5, 8, 45, 62, 83, 85, 88, 93, 96, 97, 113, 119, 122, 203, 204, 212, 238, 239, 248, 259, 350, 351, 356 transcription factor, 28–31, 33, 45, 56, 62, 68, 72, 76, 84–86, 94–97, 100, 102, 104, 108, 109, 111–113, 121–123, 133, 136, 137, 147, 201, 206, 210, 224, 228, 233, 257, 259– 267, 269–273, 275, 279, 282, 283, 347, 348, 350, 351, 359 transcriptomics, 77, 257, 260 transitivity, 450, 471 translation, 45, 62, 119, 173, 261 transmissibility, 465, 476, 477, 479 transmission, 295, 300, 302, 304, 325, 449–452, 454–457, 462–491, 493–497 transportation, 16, 467, 493, 495 tricarboxylic acid cycle, 154, 164 trophic interaction, 367–369, 373, 381, 383, 385, 386, 397, 403, 405, 444 trophic level, 366, 371, 373, 375, 380, 383, 392, 393, 401, 425, 426, 430, 435– 438, 440– 443 two-hybrid, 134–137, 140, 141, 144, 150, 206, 219 validation, 44, 45, 48, 72, 73, 75, 76, 78, 243–245, 453 vertex, 26, 102, 116–118, 120, 121, 214, 215, 280 visualization, 140–143, 147 wiring diagram, 125, 155, 156