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Power Laws in the Information Production Process: Lotkaian Informetrics

Library and Information Science Series Editor: Bert R. Boyce School of Library & Information Science Louisiana State University, Baton Rouge Recent and Forthcoming Volumes Mike Thelwall An Information Science Approach Donald Case Looking for Information Matthew Locke Saxton and John V. Richardson Understanding Reference Transactions: Turning Art Into a Science Robert M. Hayes Models for Library Management, Decision-Making, and Planning Charles T. Meadow, Bert R. Boyce, and Donald H. Kraft Text Information Retrieval Systems, Second Edition Charles T. Meadow Text Information Retrieval Systems A. J. Meadows Communicating Research V. Fronts, J. Shapiro, & V. Votskunskii Automated Information Retrieval: Theory and Methods Harold Sackman Biomedical Information Technology: Global Social Responsibilities for the Democratic Age Peter Clayton Implementation of Organizational Innovation: Studies of Academic and Research Libraries Bryce L. Allen Information Tasks: Toward a User-Centered Approach to Information Systems

Power Laws in the Information Production Process: Lotkaian Informetrics

Leo Egghe

2005

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© 2005 Elsevier Ltd. All rights reserved. This work is protected under copyright by Elsevier Ltd, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier's Rights Department in Oxford, UK: phone (+44) 1865 843830, fax (+44) 1865 853333, e-mail: [email protected]. Requests may also be completed on-line via the Elsevier homepage (http://www.elsevier.com/Iocate/permissions). In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (+44) 20 7631 5555; fax: (+44) 20 7631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of the Publisher is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier's Rights Department, at the fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. First edition 2005

ISBN:

0 12 088753 3

& The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Printed in The Netherlands.

My wife is thanking me for the many quiet evenings at the time of the writing of this book. To Ute, because she asked for it.

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PREFACE Explaining and hence understanding is one of the key characteristics of

human

beings. Explaining is making logical, mathematical deductions based on a minimum of unexplained properties, called axioms. Indeed, without axioms, one is not able to make deductions. How these axioms are selected is the only non-explainable part of the theory. In fact, different choices are possible leading to different, in themselves consistent, theories which conflict when considered together. A typical example is the construction of the different types of geometries, Euclidean and non-Euclidean geometries which are in contradiction when considered together but which have their own applications.

In this book the object of study is a two-dimensional information production process, i.e. where one has sources (e.g. journals, authors, words, ...) which produce (or have) items (e.g. respectively articles, publications, words occurring in texts, ...) and in which one considers different functions describing quantitatively the production quantities of the different sources. All functions can be reduced to one type of function, namely the size-frequency function f: such a function gives, for every n = 1, 2, 3, ..., the number f(n), being the number of sources with n items. This is the framework of study and in this framework we want to explain as many regularities that one encounters in the literature as possible, using the size-frequency function f.

As explained above, we need at least one axiom. The only axiom used in this book is that the size-frequency function f is Lotkaian, i.e. a power function of the form f(n)=C/n a , where C> 0 and a 3 0, hence also implying that the function f decreases. The name comes from its introduction into the literature by Alfred Lotka in 1926, see Lotka (1926). Based on this one assumption, one can be surprised about the enormous amounts of regularities that can be explained. In all cases the parameter a turns out to be crucial and is capable of, dependent on the different values that a can take, explaining different shapes of one phenomenon.

In this book we encounter explanations in the following directions: other informetric functions that are equivalent with Lotka's law (e.g. Zipf s law), concentration theory

viii

Power laws in the information production process: Lotkaian informetrics

(theory of inequality), fractal theory, modelling systems in which items can have multiple sources (as is the case in the system: articles written by several authors), modelling citation distributions. These models are developed from Chapter II on. Although the law of Lotka is an axiom in this book, in Chapter I we investigate what other models are capable of "explaining" Lotka's law. The only purpose for these developments is to understand why Lotka's law is choosen in this book (and not another type of size-frequency function such as, e.g., an exponential function) and hence to make the choice acceptable (although this is not needed, strictly speaking). For the same reason we also give an overview of situations where Lotka's law is encountered, being the majority of the situations (including the "new" situations as networks, including the Internet).

The author is basing himself on the publications that he has written the past 20 years on the subject but also benifits from the work of many others. To them my sincerest thanks. The author is especially grateful to Professor Ronald Rousseau (co-winner of the 2001 Derek De Solla Price Award) with whom he co-authored several papers but with whom he also had numerous long discussions (often by phone) on the different topics described in this book.

We strongly hope this book will serve the informetrics community in the sense that it shows the logical links between many (at first sight unrelated) informetrics aspects. The informetrician, having read this book, can use it each time he/she encounters a new informetric phenomenon (often in the form of a data set) in the sense that one can investigate if the phenomenon shows regularities that are (or can be) explained using the arguments given in this book. The mathematical knowledge required is limited to elementary mathematics such as first-year calculus. Other, more advanced topics, are introduced in this book.

The author is indebted to the Limburgs Universitair Centrum (LUC) and the University of Antwerp (UA) for their support in doing informetric research: in LUC, the author is chief librarian and coordinator of the research project "bibliometrics" while in UA, he is professor in the School of Library and Information Science, where he teaches the courses on informetrics and on information retrieval. The author thanks

Preface

ix

Mr. M. Pannekoeke (LUC) for the excellent typing and organization of this manuscript.

Leo Egghe Diepenbeek, Belgium Summer 2004


TABLE OF CONTENTS

Preface

vii

Table of contents

xi

Introduction

1

Chapter I Lotkaian Informetrics: An Introduction

7

1.1 Informetrics

7

1.2 What is Lotkaian informetrics ?

14

1.2.1

ThelawofLotka

14

1.2.2

Other laws that are valid in Lotkaian informetrics

19

1.3 Why Lotkaian informetrics ?

25

1.3.1

Elementary general observations

26

1.3.2

The scale-free property of the size-frequency function f

27

1.3.3

Power functions versus exponential functions for the size-frequency function f

1.3.4

32

Proof of Lotka's law based on exponential growth or based on exponential obsolescence

34

1.3.4.1 Proof of Lotka's law based on exponential growth: the Naranan model

34

1.3.4.2 Proof of Lotka's law based on exponential obsolescence: solution of a problem of Buckland

40

1.3.5

Derivation of Mandelbrot's law for random texts

42

1.3.6

"Success Breeds Success"

45

1.3.6.1 The urn model

46

1.3.6.2 General definition of SBS in general IPPs

49

1.3.6.3 Approximate solutions of the general SBS

52

1.3.6.4 Exact results on the general SBS and explanation of its real nature 55

xii


1.3.7

Entropy aspects

65

1.3.7.1 Entropy: definition and properties

66

1.3.7.2 The Principle of Least Effort (PLE) and its relation with the law of Lotka

70

1.3.7.3 The Maximum Entropy Principle (MEP)

76

1.3.7.4 The exact relation between (PLE) and (MEP)

78

1.4 Practical examples of Lotkaian informetrics

85

1.4.1

Important remark

85

1.4.2

Lotka's law in the informetrics and linguistics literature

86

1.4.3

Lotka's law in networks

87

1.4.4

Lotka's law and the number of authors per paper

90

1.4.5

Time dependence and Lotka's law

92

1.4.6

Miscellaneous examples of Lotkaian informetrics

94

1.4.7

Observations of the scale-free property of the size-frequency function f

98

Chapter II Basic Theory of Lotkaian Informetrics

101

11.1 General informetrics theory

101

II. 1.1

Generalized bibliographies: Information Production Processes (IPPs)

101

II. 1.2

General informetric functions in an IPP

104

II. 1.3

General existence theory of the size-frequency function

110

11.2 Theory of Lotkaian informetrics

114

11.2.1

Lotkaian function existence theory

114

11.2.1.1 The case pm = oo

114

11.2.1.2 The general case pm < oo

116

11.2.2 The informetric functions that are equivalent with a Lotkaian size-frequency function f

121

Table of contents

xiii

11.3 Extension of the general informetrics theory: the dual size-frequency function h

144

11.4 The place of the law of Zipf in Lotkaian informetrics

150

11.4.1

150

Definition and existence

11.4.2 Functions that are equivalent with Zipf s law

152

Chapter III Three-dimensional Lotkaian Informetrics

157

111.1 Three-dimensional informetrics

157

111.1.1 The case of two source sets and one item set

158

111.1.2 The case of one source set and two item sets

159

III. 1.3 The third case: linear three-dimensional informetrics

161

III. 1.3.1 Positive reinforcement

163

III.1.3.2 Type/Token-Taken informetrics

168

III. 1.4 General notes

172

111.2 Linear three-dimensional Lotkaian informetrics

175

111.2.1 Positive reinforcement in Lotkaian informetrics

175

111.2.2 Lotkaian Type/Token-Taken informetrics

177

Chapter IV Lotkaian Concentration Theory

187

IV. 1 Introduction

187

IV.2 Discrete concentration theory

192

IV.3 Continuous concentration theory

196

IV.3.1 General theory

196

IV.3.2 Lotkaian continuous concentration theory

199

IV.3.2.1 Lorenz curves for power laws

199

IV.3.2.2 Concentration measures for power laws

205

IV.3.3 A characterization of Price's law of concentration in terms of Lotka's law and of Zipf s law

214

xiv


IV.4 Concentration theory of linear three-dimensional informetrics

218

IV.4.1 The concentration of positively reinforced IPPs

219

IV.4.2 Concentration properties of Type/Token-Taken informetrics

226

Chapter V Lotkaian Fractal Complexity Theory

231

V.I Introduction

231

V.2 Elements of fractal theory

232

V.2.1

233

Fractal aspects of a line segment, a rectangle and a parallelepiped

V.2.2 The triadic von Koch curve and its fractal properties. Extension to general self-similar fractals V.2.3 Two general ways of expressing fractal dimensions

234 236

V.2.3.1 The Hausdorff-Besicovitch dimension

236

V.2.3.2 The box-counting dimension

239

V.3 Interpretation of Lotkaian IPPs as self-similar fractals

242

Chapter VI Lotkaian Informetrics of Systems in which Items can have Multiple Sources

VI.l Introduction

247

247

VI.2 Crediting systems and counting procedures for sources and "super sources" in IPPs where items can have multiple sources VI.2.1 Overview of crediting systems for sources

253 254

VI.2.1.1 First or senior author count

254

VI.2.1.2 Total author count

254

VI.2.1.3 Fractional author count

255

VI.2.1.4 Proportional author count

255

VI.2.1.5 Pure geometric author count

255

VI.2.1.6 Noblesse Oblige

256

VI.2.2 Crediting systems for super sources

256

Table of contents VI.2.3 Counting procedures for super sources in an IPP

xv 256

VI.2.3.1 Total counting

257

VI.2.3.2 Fractional counting

258

VI.2.3.3 Proportional counting

258

VI.2.4 Inequalities between Q T (c) and Q F (c) and consequences for the comparison of QT (c), Q F (c) and Q P (c)

261

VI.2.5 Solutions to the anomalies

266

VI.2.5.1 Partial solutions

267

VI.2.5.2 Complete solution to the encountered anomalies

269

VI.2.6 Conditional expectation results on QT (c), QF (c) and Qp (c)

270

VI.3 Construction of fractional size-frequency functions based on two dual Lotka laws

276

VI.3.1 Introduction

276

VI.3.2 A continuous attempt: z 6 R+

278

VI.3.3 A rational attempt: q G Q+

282

Chapter VII Further Applications in Lotkaian Informetrics

295

VII.l Introduction

295

VII.2 Explaining "regularities"

297

VII.2.1 The arcs at the end of a Leimkuhler curve

297

VII.2.2 A "type/token-identity" of Chen and Leimkuhler

298

V'11.3 Probabilistic explanation of the relationship between citation age and journal productivity

300

xvi


VI 1.4 General and Lotkaian theory of the distribution of author ranks in multi-authored papers

304

VII.4.1 General theory

304

VII.4.2 Modelling the author rank distribution using seeds

308

VII.4.3 Finding a seed based on alphabetical ranking of authors

310

VI 1.5 The first-citation distribution in Lotkaian informetrics

313

VII.5.1 Introduction

313

VII.5.2 Derivation of the model

317

VII.5.3 Testing of the model

320

VII.5.3.1 First example: Motylev (1981) data

320

VII.5.3.2 Second example: JACS to JACS data of Rousseau

322

VII.5.4 Extensions of the first-citation model

323

VII.6 Zipfian theory of N-grams and of N-word phrases: the Cartesian product of IPPs

326

VII.6.1 N-grams and N-word phrases

326

VII.6.2 Extension of the argument of Mandelbrot to 2-word phrases

329

VII.6.3 The rank-frequency function of N-grams and N-word phrases based on Zipf s law for N = 1

333

VII. 6.4 The size-frequency function of N-grams and N-word phrases derived from Subsection VII.6.3

347

VII.6.5 Type/Token averages nN and Type/Token-Taken averages \i"N for N-grams and N-word phrases

352

Appendix

365

Appendix I

365

Appendix II

370

Table of contents xvii

Appendix III Statistical determination of the parameters in the lawofLotka

372

A.III. 1 Statement of the problem

372

A.III.2 The problem of incomplete data (samples) and Lotkaian informetrics

373

A.III.3 The difference between the continuous Lotka function and the discrete Lotka function

378

A.III.4 Statistical determination of the parameters K, a, nmax in the discrete Lotka function K/na, n = 1 ,...,nmax

386

A.III.4.1 Quick and Dirty methods

387

A.III.4.2 Linear Least Squares method

388

A.III.4.3 Maximum Likelihood Estimating method

390

A.III.5 General remarks

393

A.III.5.1 Fitting Zipf s function

393

A.III.5.2 The estimation of pm and nmax

394

A.III.5.3 Fitting derived functions such as Price's law

394

A.III.5.4 Goodness-of-fit tests

395

Bibliography

397

Subject Index

423


INTRODUCTION The most facinating aspect of informetrics is the study of what we could call twodimensional informetrics. In this discipline one considers sources (e.g. journals, authors, ...) and items (being produced by a source - e.g. articles) and their interrelations. By this we mean the description of the link that exists between sources and items. Without the description of this link we would have two times a one dimensional informetrics study, one for the sources and one for the items. Essentially in two-dimensional informetrics the link between sources and items is described by two possible functions: a size-frequency function f and a rank-frequency function g. Although one function can be derived from the other, they are different descriptions of two-dimensionality. A size-frequency function f describes the number f(n) of sources with n = 1, 2, 3, ... items while a rank-frequency function g describes the number g(r) of items in the source on rank r = 1, 2, 3, ... (where the sources are ranked in decreasing order of the number of items they contain (or produce)). So, in essence, in f and g, the role of sources and items are interchanged. This is called duality, hence f and g can be considered as dual functions.

Rank-frequency functions are well-known in the literature, especially in the economics and linguistics literature, where one usually considers Pareto's law and Zipf s law, respectively, being power laws. Less encountered in the literature (except in information sciences) is the size-frequency function. If studied, one supposes in most cases also a power law for such a function, i.e. a function of the type f(n)= C / n a with a > 0. Such a function is then called the law of Lotka referring to its introduction in the informetrics literature in 1926, see Lotka (1926). The law of Lotka gives rise to a variety of derived results in informetrics, the description of them being the subject of this book. That we choose a size-frequency function as the main studyobject is explained e.g. by its simplicity in formulation (in the discrete setting simpler than a rank-frequency function since the latter uses ranks which have been derived from the "sizes" n but also in the continuous setting, where sizes and ranks are taken in an interval, the formulation of the size-frequency function is more appealing and direct). A size-frequency function also allows for a study of fractional quantities (see Chapter VI), needed e.g. in the description of two-dimensional informetrics in which

2


items can have multiple sources (as e.g. is the case with articles that can be written by several authors - incidently the original framework in which the law of Lotka was formulated in 1926). Furthermore, in this book we restrict ourselves to size-frequency functions that are power functions, i.e. Lotkaian informetrics. In Chapter I (a complete overview of the chapters will follow in the sequel) several arguments are given why power functions have to be preferred above other functions (e.g. exponential functions): a major reason is that power functions satisfy the so-called "scale-free" property meaning that the multiplication of the size n with a constant C > 0 does not change the type of power function, a property that is characterizing power functions. This is an important property since direct comparisons of results (using the same function) can be made whether we work with small numbers n or high numbers n. Also scale-free functions are the only ones capable of describing the self-similar fractal aspects of the informetric system under study (Chapter V). Furthermore in Chapter II we show that power-type size-frequency functions comprise all power-type rank-frequency functions (such as Zipfs law) but also other rank-frequency functions such as Mandelbrot's function and even exponential rank-frequency functions. This is an important theoretical advantage of the use of Lotka's law above any rank-frequency law: the class of scale-free size-frequency functions is strictly wider than the class of scale-free rank-frequency functions. This means that Lotkaian informetrics describes results which have a wider application than e.g. the study of Zipfian informetrics which is included in the Lokaian one.

In Chapter I we first explain the ingredients of informetrics and of Lotkaian informetrics. The use of Lotka's law for the size-frequency function could be considered here as an axiom in the sense that it is not explained but assumed - hereby allowing for other informetrics theories based on other size-frequency functions (a bit comparable with the establishment of different types of Euclidean and non-Euclidean geometries). Nevertheless we give, in Chapter I, several "explanations" of Lotka's law. These explanations are based, of course, on other assumptions that are not explained but in this way links between different regularities are given. So we present a proof of Lotka's law based on exponential growth, as proved by Naranan (1970) and a proof of Lotka's law based on exponential aging solving hereby a problem of

Introduction 3 Buckland (1972). Mandelbrot's law, being equivalent with Lotka's law is derived for random texts. The relation between Lotka's law and the "Success Breeds Success" phenomenon is presented. Finally, we give a description of the Principle of Least Effort (PLE) and the Maximum Entropy Principle (MEP) and show how Lotka's law can be derived from the (PLE) but not from the (MEP). To fully understand this remarkable result a characterization of (MEP) in terms of (PLE) and the newly defined Principle of Most Effort (PME) (defined in Egghe and Lafouge (2004)) is given. The extensive introductory Chapter I is then ended with an overview of practical examples of Lotkaian informetrics (including the recently found power-type laws in networks). Chapter II presents the basic theory of Lotkaian informetrics: describing the system of sources and items as information production processes (IPPs) and the general informetric functions to be studied: size- and rank-frequency functions. Given A, the total number of items and T the total number of sources, we give necessary and sufficient conditions for the existence of a size-frequency function giving these numbers A and T. Based on this we determine the existence of the corresponding Lotkaian functions. Then we determine the functions that are equivalent with the law of Lotka, for the diverse values of a > 0. In this connection we find the laws of Mandelbrot, Leimkuhler and Bradford (the latter one being a group-free version of the classical law of Bradford) and determine the relations between the parameters of these laws. We further prove that Zipfs law is, at least for special values of C, also equivalent with the law of Lotka. Also in Chapter II we define the dual size-frequency function h and prove its properties. This function is used later (in Chapter VI) to determine the number h(s) of authors with a fractional score s = 1/j per paper, j = 1, 2, 3,..., where j is the number of authors of a paper. This shows that the dual framework of IPPs and of size-frequency functions is capable to handle IPPs where items can be produced by multiple sources. Chapter III is devoted to the relatively new and intricate problem of three-dimensional informetrics. Especially linear three-dimensional informetrics is studied in general and in the Lotkaian case. Here two types of studies are given. One on positive reinforcement, i.e. where sources produce items and where these items become

4


sources that produce again items, and one on Type/Token-Taken (TTT) informetrics in which not only sources and items (i.e. in the linguistical terminology: type and token) are considered but also the use (i.e. taken) that is made of these items. In both cases, concrete formulae for the size- and rank-frequency functions are given, based on the ones of the composing IPPs and consequences are given supposing Lotkaian informetrics. In both cases we also study the new averages of these cases of linear three-dimensional informetrics and prove that in all cases the average increases. Chapter IV is devoted to the study of concentration theory of Lotkaian informetrics, i.e. the determination of measures for the inequality between sources (with respect to the number of items they contain). Especially in Lotkaian informetrics this inequality is apparent due to the skewness of the Lotka function. Inspired by methodologies coming from econometrics we define the continuous Lorenz curves, which have the property that, the higher the curve, the more unequal the studied situation is. We also calculate these curves in the case of Lotkaian informetrics and prove that these curves increase in a (i.e. inequality increases with a). Based on these curves we are then able to present several good measures of concentration (i.e. measures of inequality) for Lotkaian systems. Also the concentration of the linear three-dimensional systems, studied in Chapter III, is determined: in a case of positive reinforcement the concentration is ruled by a theorem of Fellman (1976) which is reproved here; in the case of TTT we generally show that the concentration decreases. In this Chapter we also consider the so-called law of Price which we prove to be equivalent with Zipf s law with exponent 1, hence with Lotka's law with a = 2 and where the constant C takes a special value, a fact that was already found to be true (experimentally) in Allison, Price, Griffith, Moravcsik and Stewart (1976).

Chapter V gives the already mentioned interpretation of a Lotkaian system as a selfsimilar fractal, after giving a general introduction to fractal theory. The formula for the fractal dimension Ds of such a system (a measure of its complexity) then leads, via the theory of Naranan (1970), given in Chapter I, to the important relation: Ds = a - 1 .

Introduction 5 Chapter VI is completely devoted to the general theory of systems in which items can have multiple sources and to the Lotkaian involvement in this. We start with a general overview of crediting systems for sources, such as the total, fractional, proportional,... counting methods. We show the robustness of Lotka's law when going from singlesource items systems to multiple-source items systems when one uses the total counting system. These different counting procedures are also defined and applied to so-called "super sources", e.g. countries adding scores of the different authors (sources) in these countries. Some paradoxes in the relative scores between these procedures are highlighted and remedies given. Chapter VI then continues with the construction of fractional size-frequency functions based on the Lotka laws f and its dual (cf. Chapter II) h. The surprising result, using convolution theory, being irregularly shaped size-frequency functions over some rational scores is obtained by applying "ordinary" decreasing Lotka functions f and h. The closeness with practical data is remarkable.

The final Chapter VII gives further applications in Lotkaian informetrics. We alert the reader that not all regularities to be explained are of the informetric type: this is the case with the arcs at the end of a Leimkuhler curve and with a "type/token-identity" of Chen and Leimkuhler (1989). Based on the Central Limit Theorem in probability theory an explanation is given of the relationship between citation age and journal productivity, experimentally established by Wallace (1986), where part of the graph is explained using Lotka's law. Further, we give the general and Lotkaian theory of the distribution of author ranks in the byline of multi-authored papers. Lotkaian informetrics is also able to model the cumulative first-citation distribution where Lotka's a is the key parameter which decides on a concave shape (l < a < 2) or an S-shape (a > 2) of this distribution. Finally the size- and rank-frequency distributions of N-grams and N-word phrases are determined based on the law of Zipf (which is part of Lotkaian informetrics) for N = 1. We could also speak about a theory for the Cartesian product of IPPs. The Appendix is mainly devoted to the statistical determination of the parameters in the Lotka function. We first determine the influence of incomplete data (samples) on the law of Lotka. We show that Lotka's law is the only one allowing for the same size-

6


frequency distribution for the complete and incomplete data, making it the only one treatable in sampling problems. Then we describe the difference between the continuous Lotka functions and the discrete ones, when they are used to model the same data and we determine the relations between the parameters. Then we present some statistical methods for the determination of the parameters: after some "quick and dirty" methods we give the linear least square method and the maximum likelihood estimating method.

I

LOTKAIAN INFORMETRICS : AN INTRODUCTION

I.I INFORMETRICS The concept of informetrics is well-known nowadays and could be defined as the science dealing with the quantitative aspects of information. This is the widest possible definition comprising mathematical and statistical treatment of diverse forms of information: books, articles, journals, references and citations, libraries and other information centers, research output, collaboration and transport (e.g. in networks such as intranets or the internet). Although there might be different opinions on this, informetrics could be considered (as we will do here) as comprising other disciplines such as bibliometrics or scientometrics. Bibliometrics can be defined as the quantitative study of pieces of literature as they are reflected in bibliographies (White and McCain (1989)) or by the well-known definition of Pritchard (1969): the application of mathematical and statistical methods to books and other media of communication (see also Narin and Moll (1977) and Tague-Sutcliffe (1994)). Scientometrics, coined in Nalimov and Mul'cenko (1969) as "naukometrija", deals with quantitative aspects of science, including research evaluation and science policy.

The term informetrics, we believe introduced in 1979 by Blackert and Siegel (1979) and by Nacke (1979), gained popularity by the organization of the international informetrics conferences in 1987 (see Egghe and Rousseau (1988, 1990b)) and by the foundation (during the fourth international informetrics conference in 1993) of the ISSI, the International Society for Scientometrics and Informetrics, hereby also recognizing the importance of the term scientometrics, mainly because of the existence of the important journal with the same name. It is not the purpose to repeat in this book all historic facts on the science informetrics since this has been covered many times in an excellent way in publications as White and McCain

8


(1989), Ikpaahindi (1985), Lawani (1981), Tague-Sutcliffe (1994), Brookes (1990) and the more recent, very comprehensive (almost encyclopaedic) Wilson (1999).

Since, however, we intend to provide mathematical foundations for a part of informetrics (called Lotkaian informetrics, of course explained further on) we will provide in this overview a concrete description of the concept of generalized bibliography (or information production process) as well as the standard definitions of informetric functions, both as a concept and as they appear in the literature (then called laws). They form the basis of the informetric theory and will be formalized in Chapter II.

The main object of study in informetrics is the generalized bibliography, also called (e.g. in Egghe (1990a)) an "information production process" (IPP). The most classical example is, indeed, a bibliography (on a certain subject) where one has a collection of articles dealing with this subject. Of course, articles are published in journals and this is the basic aspect of IPPs: in the example of a classical bibliography, journals can be considered as sources that "produce" items, i.e. the articles collected in the bibliography. The point is that, in informetrics, one can provide several other examples of sources, containing items. Indeed one can consider the publications (articles) of an author also as a source (author)-item (publication) relationship: an author "produces" a publication. Even an article (being an item in the previous examples) can become a source e.g. "producing" references or citations as items. In a library, books (as sources) are the "producers" of loans (each time a book is borrowed this is an item belonging to the source, being the book itself). Although this example might seem more abstract than the previous ones it is a very natural example and of the same nature as (although completely different from) the following example. In quantitative linguistics (for a basic reference, see Herdan (I960)) one considers texts as (in our terminology) IPPs where words are considered as sources and their use in the text (i.e. each time a word appears in the text) is considered as an item. There one uses the terminology "Type/Token"-relationship for what we call here the "Source/Item"-relationship (see also Chen and Leimkuhler (1989)). In this book we will use both versions of the terminology: mainly source/item, since this is classical in informetrics; type/token being interesting descriptions of the same phenomenon and sometimes used, e.g. where it is more convenient such as in Chapter III. We are convinced this will not confuse the reader (again: there is no difference: source/item = type/token) and in this way we underline the fact that our

Lotkaian informetrics: an introduction

9

framework is applicable outside informetrics (although linguistics is a neighbor discipline of information science!).

Really outside informetrics we are still able to find the same framework. We can give the example in econometrics of workers or employees (as sources) in relation with their productivity (as items) (Theil (1967)). Productivity can be expressed by the number of produced objects by these sources or in terms of profits (amount of money earned by these sources). Even in demography one can consider cities and villages (as sources) in relation to their populations (each member being an item).

So, in general, we can define an IPP as a triple (S,I,F) where S is the set of sources, I the set of items and F a function indicating which item i e l belongs to which source seS. In this sense we can talk about two-dimensional informetrics studying sources/items and their interrelation by means of F. Two-dimensional informetrics is hence more (higher) than two times a one-dimensional informetrics theory developed separately on the sources (e.g. number of sources) and on the items (e.g. number of items). The function F can be considered as a relation F: S —> I or as a function F: S —> 21 (the set of all subsets of I) where, for each s e S, F ( s ) d , the subset of I containing the items that are produced by (that belong to) source s. The classical way of thinking, as we will do in this first chapter, is limiting the sets S and I to finite discrete sets (e.g. finite subsets of N , the set of natural numbers, so that we can count them). For the basic theory, however, in Chapter II, we will express the fact that IPPs usually contain many sources and items, by using continuous sets for S and I (such as intervals [a,b] c R + , the positive real numbers).

The reader might remark that this common framework is nice but does it lead us somewhere, i.e. is there a reason to formulate these objects in a common way? The answer is yes, for several reasons. First of all there is the challenge in itself to detect and define common tools or frameworks (such as IPPs) among these different sciences. Once this is done we can then elaborate these tools in a unified way e.g. by defining common measures and functions (such as distribution functions - see further, but these are not the only examples). An intriguing benefit of this approach would be if we detect or prove that some of these measures or functions have the same form in the different "-metrics" sciences! While leaving this

10


"speculation" for what it is (for the moment, but watch the sequel!) we will now give some common functions and measures that can be defined in these IPPs.

So we suppose that we have an IPP (S,I,F), where - for the time being - S and I are finite sets. As defined above, for each se S, F(s) is the set (subset of I) containing the items that belong to (or are produced by) s. Some general quantities can be defined (note that this also helps us in the definition of standards in informetrics (and beyond), the need of it being advocated e.g. in Glanzel (1996) - see also Rousseau (2002a)). Knowing (S,I,F) we can determine, for every n s N (i.e. for every number n = 1,2,3,...) the number f(n) of sources with n items (i.e. having or producing n items). The function f is called the size-frequency function of (S,I,F) and is the basic informetric function in this book. Both its simplicity (see next section) and its power to determine quantitative regularities in IPPs will be highlighted during the elaboration of this book. Note that this function f can be determined by, simply, counting for each source the number of items its contains and then, for each ne N, counting the number of sources with this number n of items. Note that T = ^ f (n) denotes the total number of sources in the IPP and that A = ^ n f (n) denotes the total number of items in the IPP. Hence

A

E nf («)

(L1)

H= — = H h

T

f n

£( > n

is the average number of items per source in this IPP, an important measure. Here E means n

the addition (sum) for all existing values of n, usually ne {1,2,3,...,nmax} where nmaX is the maximal production of a source in this IPP. We note here the (intuitively clear) fact that not all values in {l,2,...,nmax} are effectively encountered: usually small values (1,2,3,...) always occur and in high quantities (in other words: there are many low productive sources) but higher productions are rare: high n values do not occur or, if they occur, not many sources (usually 1) have this number of items. This intuitive description is basic for the rest of the book. We will see (and model) the fact that a high number of sources have a low number of items and that a low number of sources have a high number of items. Stated in terms of f (and in an intuitive way): f is a (mainly) decreasing function of n (and, of course, f(n)>0 for all n). This simple property is the start of a limiting procedure for possible functions f which


11

will lead us (in this chapter) to the arguments why we will study "Lotkaian" informetrics from Chapter II on: Lotkaian refers to a certain special, simple, type of function f, to be introduced in the next section. The inequality in terms of the source production (number of items per source), as described above, will be explicitely studied (by defining so-called inequality measures) in Chapter IV.

Let us now continue to define more types of functions and measures in the IPP (S,I,F). First we need to define a ranking in the set of sources. This is done as follows: the source with the highest number of items receives rank 1, the source with the second highest number of items receives rank 2 and so on. Rankings for sources with an equal number of items are given consecutively where it is of no importance which source receives which rank. Example: Suppose S = {s 1 ,s 2 ,s 3 ,s 4 ,s 5 ,s 6 } and let Sj have 3 items, s2 have 5 items, s3 and s4 have 1 item and s5 and s6 have 2 items. Then s2 receives rank 1, s, receives rank 2, s5 receives rank 3 and s6 rank 4 (or s5 and s6 interchanged), s3 receives rank 5 and s4 rank 6 (again s3 and s4 can be interchanged). Stated mathematically in the notation of our IPP (S,I,F), we have, if r(s) denotes the rank of source s e S, then for all s,s' e S for which |F(s)| * |F(s')|:

r(s) 0) and the exponent a will determine the "degree" of this inequality among the sources. To f(l) determine the fraction —— of sources with 1 item, for general a , we can proceed as follows: again we have


17

so

c

-

f ( 1 )

!

-

a 13) M

T

T

|J

I

f^J_ n=l

n

which can be determined from the tables of the so-called Riemann-zeta function

n=]

n

We again refer to the Appendix III for an approximative formula and Table for (1.14), given a , and for a statistical method to estimate a itself. In the theory, to be developed from Chapter II on, we will see that, in the continuous setting, it will be much easier to determine C and a , given T (the total number of sources) and A (the total number of items). The Riemann-zeta function is not needed there. We can say that Chapter II will present a method of "mathematical fitting" of the Lotka function, complementing the statistical fitting, given in the Appendix III.

The law of Lotka is the simplest among the informetric laws (to be formulated in the sequel of Section 1.2). Yet it is remarkable that its historical formulation in Lotka (1926) was in terms of authors (as sources) and their publications (as items). Why is this remarkable? Of the many examples of IPPs, formulated in Section I.I, it is the only one where items can have multiple sources, i.e. in terms of the historical formulation, articles can have multiple authors. In all the other examples an item is produced by a single source: an article by 1 journal, a reference is produced by 1 article or book (the same for a citation), a loan refers to 1 book, a word token in texts obviously comes from 1 word type, an inhabitant is linked to 1 city or village, a produced item (in economics) (e.g. money earned) to 1 employee (although in this case an example of a multiple source item can be given in the sense that a product can be produced by the collaboration of several employees).

18


Of course, in a purely dual setting (section I.I) any IPP can, if necessary via its dual IPP (see section I.I), be considered as a system in which items can have multiple sources, since in the original IPP, sources usually have more than 1 item (and then interchange the names "source" and "item"). Especially the IPPs defined via references or citations can be considered in this way.

In any case, the author-article relationship is, because of the above remark, rather unique in informetrics (contrary to the statements in Fedorowicz (1982a) and White and McCain (1989)). This is the reason why we added a special Chapter (VI) solely devoted to the aspect of this multiple source possibility. One of the main issues in this setting is the question how to count the number of items per source. Indeed, suppose we have an item with 3 sources, e.g. an article co-authored by 3 authors. Will each author receive a (production) credit of 1 (total count) or only of — (fractional count, see also De Solla Price (1981))? Of course, the fractional counting method is in better agreement with the dual framework: "if there are n sources per item then there are, in a dual vision, — items per source" (n e N). n One can even consider straight counting which gives a credit of 1 to the first author and 0 to the other authors (Cole and Cole (1973)). A variant of this is to give a credit of 1 to the senior author (usually the last author, at least in those disciplines where the senior author is identified) and 0 to the other authors : noblesse oblige, see Zuckerman (1968) or see Subsection VI.2.1.6. For more on these and other counting methods we refer the reader to Egghe, Rousseau and Van Hooydonk (2000) and to section VI.2. Lotka (1926) used the latter method (senior author count), hence treating papers as single authored and hence also circumventing the problem of multiple authorship. Potter (1981) thinks to have an explanation for Lotka's choice of counting only senior authors. First, in Lotka's Chemical Abstracts data, only the first four authors are indexed. However, the second, third and fourth author only receive a "see" reference to the first author and from this it is a lot of work to compile author productivity data based on total counts. Potter then assumes that Lotka avoided this amount of work and therefore only counted the senior author. Yet it remains an interesting problem (addressed in Chapter VI) why Lotka's law is so widespread both in IPPs where items have only one source and in IPPs where items can have multiple sources (see Section 1.4). Even Lotka himself mentions the "wide range of applicability of the power law to a variety of


19

phenomena" and even refers to econometrics and biometrics and to aspects of concentration (inequality), see Chapter IV. Also interesting is: how do we go from laws in IPPs where we have single source items to laws in IPPs where we have multiple source items, both in the total and fractional counting system? We will see that Lotka's law will be the key function in both explanations (Chapter VI).

1.2.2 Other laws that are valid in Lotkaian informetrics

In a purely formal sense it is impossible, in the discrete setting, to produce formulae for functions (such as g, G, a p or S p introduced in Section I.I) that are equivalent (or at least imply) Lotka's law. The reason is that discrete sums (as occurring in (1.3), (1.4) and (1.6)) cannot be evaluated. A real powerful Lotkaian informetrics theory can only be produced in a continuous setting, as will be done from Chapter II on. Now we will limit ourselves to the definition of some historical laws which we will prove in Chapter II play a role in (continuous) Lotkaian informetrics; otherwise stated: laws which are compatible with Lotkaian informetrics.

Related to the function g (1.3) we have two versions. First there is the law of Zipf:

g(0 = 7'

(L15)

where D and P are positive constants. In this setting, the number of items in the source on rank r is a decreasing power function of r. In this sense, this law is equal (as a mathematical function) to the one of Lotka (1.7) but it will be seen in Chapter II that the exponents are not the same and, furthermore, that Lotka's law embraces a wider class of rank-frequency functions, named after B.B. Mandelbrot, the inventor of fractal theory (complexity theory see also Chapter V or Mandelbrot (1967, 1977a,b)): the law of Mandelbrot:

g(r)=

(T7kr

(L16)

20


where E > 0, F and p are constants (as will be seen in Chapter II, [3 can be negative but then F < 0, keeping g decreasing).

Of course, dividing by F p in the nominator and the denominator of (1.16) leads to the following trivial equivalent variant of (1.16)

8

, ,y ' N(t 0 )

(1.38)

is the fraction of the sources (at t 0 ) that started (were "born") in the time interval ]0,t]. Hence

^ 4 N(t 0 ) is the density (at t 0 ) of the sources that started at t itself: indeed

r.N£h = N(t)-N(0) °N(t0) N(t0)

J

(L39)

36


being equal to (1.38). So, taking t = t0 — i,

N

V/)

(1.41)

N(t0) is the density of sources that are (at t0) x time units old. (1.41) equals

c,(lna,)ar c,a;»

= (lna,)ar'

(1.42)

Since this is independent of t 0 , (1.42) represents the overall age density. To go from the variable x (age of source) to the variable p (number of items in a source) we use the cumulative distribution of sources that are less than or equal to x time units old:

f(lna,)ar T 'dx'

We now use (1.37) twice:

P(X) =

C a

2 2

implies

-j2. = c2(lna2)a^

and

In -2- Uxlna,

(1.43)

Lotkaian informetrics: an introduction 37

which yields

dT

= W = T^ c 2 (lna 2 ja 2

(L44)

plna 2

and In P-' x=- ^ ~ lna 2

(1.45)

(1.44) and (1.45) in (1.43) yields (replacing the dummy variable x' by the dummy variable p ' )

1

'

F(p)= K

'

J o ina 2 p '

'

rÂvi^^ Jo I n a 2 [ c 2

p'

F(p) = / ; ^ - c ^ p ^ i d p '

(L46)

being the cumulative distribution function of sources with less than or equal to p items. Hence, the size-frequency distribution f(p) is nothing else than F'(p). Hence

f( p ) = p . c 2 t a - p l ma,

b

"

(1.47)

, hence Lotka's law with exponent a = 1+ ]na i

lna2

(I4g)

38


So we have shown that Lotka's law can be derived from exponential growth and furthermore, the exponent a (Lotka's exponent - see formula (1.28)) is related to this (double) exponential growth via formula (1.48) which will play a crucial role in Lotkaian informetrics: (1.48) will be basic for the fractal description of Lotkaian IPPs as self-similar fractals (see Chapter V). This fact was not mentioned in Naranan and we think it is revealed here.

We now present the same argument but treating time as a discrete variable (t = 0,1,2,3,-

) and

we will show that the same result is valid, hence also showing that the differential calculus of Naranan (giving rise to the dispute in Hubert (1976)) is not needed: an elementary argument of counting provides the same result, showing again that the criticism of Hubert (1976) is not correct. So we suppose we have (1.36) and (1.37) for discrete t. Let t e N be fixed but arbitrary. At time t there is a fraction of — N(t)

sources that exist i or more time units (e.g. a

.»w year), hence with p or more items, where p = c2a2, hence i = —s-2-l. Hence the fraction lna2 N(t-i)

^Teq

, 4) N(t-i) N(t)

=

=

c,a,

'

1

41 lna.

One can write that


G

( p ) = (-) ta * 2

39

d- 49 )

which is the cumulative fraction of sources with p or more items, i.e. G = 1-F with F as in the previous (continuous) argument. Hence the size-frequency function f(p) relates to G as

G(p)= P f ( p ' ) d p ' .

Hence we have

f(p) = -G l (p)

c 2 lna 2 ( p

, hence Lotka's law C/p" (identical with (1.47)) with a as in (1.48). This shows the validity of Naranan's argument. We rephrase this important result as a theorem.

Theorem 1.3.4.1.1: In any IPP, let the sources grow exponentially in time with rate a, and let the items in the sources grow exponentially in time with fixed rate a 2 , then this implies that the size-frequency function of this IPP is of power type (hence Lotkaian) f(p) = C/p a where

a

_1

=

]naL lna2

the quotient of the logarithms of the growth rate of the sources and the one of the items in the sources. The depth of this result will be explored further on in this book.

We have the following trivial but interesting corollary.

40


Corollary 1.3.4.1.2: If the exponential growth rate of the sources equals the exponential growth rate of the items in the sources, then we have Lotkaian informetrics with Lotka exponent a = 2.

1.3.4.2 Proof of Lotka's law based on exponential obsolescence: solution of a problem of Buckland The problem described here was originally formulated in Buckland (1972) and re-formulated and solved in Egghe (2004a).

Obsolescence is usually expressed by the decline in time of the use of a document and use is, in most cases, expressed as "cited". Here time goes to the past and one talks about synchronous obsolescence as opposed to diachronous obsolescence where one studies the citation of a document (or a group of documents) after its publication (cf. Stinson (1981), Stinson and Lancaster (1987)). Here we will limit ourselves to synchronous obsolescence and simply talk about obsolescence.

There are at least two ways of considering obsolescence: the decline of the use of an article the older it is but also, as a consequence of it, the decline of the use of a journal (in which this article is published). So, in general, we can study obsolescence of items and obsolescence of sources. Buckland (1972) uses the terminology "obsolescence" for the former and "scattering" for the latter and poses the question:

Problem of Buckland: What is the relation between obsolescence and scattering? In our terminology: what is the relation between obsolescence of the items and obsolescence of the sources?

We think this formulation is incomplete, for the following reasons. Obsolescence of sources is an incomplete way of describing scattering, the latter usually being described as the distribution of items over sources, e.g. the size-frequency distribution in two-dimensional informetrics. Secondly, the obsolescence distribution of sources does not follow from the obsolescence distribution of the items since the disappearance of an item in time (in the past when the item is not used anymore) does not necessarily lead to a disappearance of the source


41

which contains this item (since the source can still contain other items that are still used) [This is related to the success-breeds-success mechanism to be described in Subsection 1.3.6]. So, describing obsolescence essentially requires two distribution functions: the one of the itemaging and the one of the source-aging and hence we can talk about two-dimensional obsolescence. In fact this is in line with the graphs produced in Buekland (1972), where this type of two-dimensional obsolescence is depicted. With two-dimensional obsolescence given, we are in a logical position to ask for the relation with two-dimensional informetrics (scattering) as e.g. expressed by the size-frequency distribution.

Therefore we think that the problem of Buekland can be reformulated as follows (and inspired by Buekland (1972)).

Reformulation of the problem of Buekland: What is the relation between two-dimensional obsolescence and two-dimensional scattering (as e.g. expressed by the size-frequency function)?

In Egghe (2004a) this problem is solved as follows

1.

We suppose we have given an exponentially decreasing obsolescence distribution for the age of the items (citations) (to the past) and an exponentially decreasing obsolescence distribution for the age of the sources (that contain these items, e.g. the journals that contain these cited articles); also here, time is going to the past. Note that (see e.g. Egghe and Rao (1992a)) exponential aging distributions are the basic functions to describe obsolescence (i.e. aging).

2.

We reverse the time by applying the transformation t' = —t, so we transform the "past" into the "future". Looking at time this way, exponentially decreasing obsolescence distributions become exponentially increasing growth distributions, hence distributions as given by Naranan and described in Subsection 1.3.4.1: exponential growth of both the sources and the items.

3.

It then follows from Naranan's argument (Subsection 1.3.4.1) that we have a sizefrequency distribution of power-law-type, hence Lotka's law.

Hence we have the following Theorem, being an answer to the problem of Buekland.

42


Theorem 1.3.4.2.1 (Egghe (2004a)): Let us have a two-dimensional obsolescence situation in which the number of sources decreases exponentially in time and where the number of items also decreases exponentially in time (with the same aging rate in each source), then we have an IPP where the sizefrequency function is Lotkaian.

We refer to Egghe (2004a) for more details. 1.3.5 Derivation of Mandelbrot's law for random texts

We will now consider the special type of IPP where sources are word types and items are word tokens as occurring in linguistics (cf. the examples of IPPs in Section I.I). We will derive the rank-frequency function g, as introduced in Section I.I and show that the law of Mandelbrot (1.16), introduced in Section 1.2, applies. There we indicated (but its proof will be given in Chapter II) that the law of Mandelbrot is equivalent with Lotka's law, hence, in this way, a new explanation of Lotka's law is given (albeit for random texts). The argument will result in a formula for the exponent (3 in Mandelbrot's law, similar to the result obtained in the previous subsection. In fact, Chapter II will fully explain the relation between both exponents ( a of Lotka's law and [3 of Mandelbrot's law) and will show that both results are the same.

The argument given by Mandelbrot is a support for the Naranan argument and, furthermore, Mandelbrot's reasoning will be extended, in Chapter VII, for the derivation of the rankfrequency distribution of N-word phrases. We give Mandelbrot's argument as indicated in Mandelbrot (1977b) (cf. also Nicolis, Nicolis and Nicolis (1989) or Li (1992)) and clarified in Egghe and Rousseau (1990a).

Let our alphabet (including all necessary numbers or signs) consist of N "letters". Consider a text as consisting of letters and blanks. Every letter or blank fills up the spaces in the text. We assume (greatly simplified situation) that every letter has an equal chance of being used. Let p be this probability. Therefore, since there are also blanks in the text:


43

p = P(letter) 1, when, from t + 1 on, many new sources arrive (probability a(t + 1),...), |i(t) will start diminishing since these new sources (temporarily), have only 1 item).

Note that, for all t, E, (X(t)) = E(X(t)), the expectation of X(t). In terms of Fig. 1.3.2: E, (X(t)) = E(X(t)) gives the (conditional) expectation of X(t) only based on the knowledge at t = 1. Notice how fine the conditional expectations Et(X(t + 1)) are: here we only average points on level t + 1 which have a common splitting point at level t, hence giving finer estimates for X(t + 1) than the rough E(X(t +1)).

The sequence X(t), t = 1,2,... together with the conditional expectations Et(X(t + 1)), t = 1,2,... form a stochastic process (as defined in probability theory - see e.g. Neveu (1975), Egghe (1984)): typical for such a process is that it is not possible (although tried in Subsection 1.3.6.2 with approximations), for any t = 1,2,... to determine X(t + 1) from X(t) (this would mean that the gambler can predict the outcome!) but that from X(t) one can determine Et(X(t + 1)). This is what SBS is all about, in an exact sense. It is now clear how to proceed: the theory of stochastic processes is available (again, see Neveu (1975), Egghe (1984)) and the results are hence applicable to SBS of IPPs. But first, we must determine the exact form of Et (X(t +1)) for the different informetric processes:

1.

T(t), the total number of sources at t

2.

\i (t), the average number of items per source, being

*

( 1 )

^

as we have indentified discrete time t with the total number of items. 3.

for every n = 1,2

t fixed, f(t,n), the number of sources with n items.

58


Recall that we work with the general SBS principle as defined in (i), (ii) in Subsection 1.3.6.2.

Theorem 1.3.6.4.1 (Egghe and Rousseau (1996a)): For every t = 1,2,...

1.

3.

E,(T(t + l)) = T(t) + a(t)

(1.76)

For n= 1.

E t (f(t + l,l)) = f(t,l) + a(t)-x(t,l)(l-a(t))

(L79)

E t (f(t + l,n)) = f(t,n) + (l-a(t))(x(t,n-l)-x(t,n))

(1.80)

E t (f(t + l,t + l)) = (l-a(t))x(t,t)

(1.81)

Forn = 2,...,t

For n = t+1

Proof:

1.

Et(T(t + 1)) = a(t)(T(t) + 1) + (1 - a(t))T(t) = T(t) + a(t)

by the SBS definition. Note that the values x(t,n) in the second part of the SBS


59

definition are not needed: indeed T(t) is determined by the fact that the new item (the (t + l)th) will go to a new or an already existing source (respectively with probabilities a(t) and 1 - a(t)) and not by which already existing source (i.e. not by the number of items that are already in this existing source at t). The same remark goes for [i since H(t) = t/T(t).

2.

E, W . + .))-E I

,

J5 '

>

a(t)

l-a(t)

The other formula follows from \i (t) = t/T(t).

3.

For the f(t + l,n), n = 1,2,...,t + 1, obviously, the x(t,n) are involved. For n = 1, we have

Et(f(t + 1,1)) = a(t)(f(t,l) + 1) + (1 - a(t))[(f(t,l) - l)x(t,l) + (1 - x(t,l))f(t,l)]

= f(t,l) + a(t)-x(t,l)(l-a(t)) For n = 2,...,t, we have

Et(f(t + l,n)) = a(t)f(t,n) + (1 - a(t))[x(t,n)(f(t,n) - 1) + x(t,n - l)(f(t,n) + 1) + (1 - x(t,n) -x(t,n-l))f(t,n)]

= f(t,n) + (1 - a(t))(x(t,n - 1) - x(t,n))

Finally, for n = t + 1, the formula E(f(t + l,t + 1)) = (1 - a(t))x(t,t) is clear.

D

60


These are the basic SB S equations for the stochastic processes T(t), u (t) and f(t,n), n = 1,... ,t. Let, again, X(t) be any of such processes. We can apply the following definitions and results of stochastic processes to our SBS formulae.

Definitions 1.3.6.4.2:

1.

A stochastic process X(t) is called a martingale if

E,(X(t + l)) = X(t)

(1.82)

for all t e N .

2.

A stochastic process X(t) is called a submartingale if

E t (x(t + l))>X(t)

(1.83)

for all t e N

3.

A stochastic process X(t) is called a supermartingale if

E,(x(t + l))<X(t)

(1.84)

for all t 6 N. In terms of the gambler's fortune, we have a martingale if the game is fair in the sense that the gambler can expect to keep his/her capital after gambling. In case of a submartingale, the gambler is expected to win. Casinos will not allow this: at least to cover their expenses, games are usually supermartingales in which a gambler is expected to loose. We have the following results on (sub-) (super-) martingales, cf. Neveu (1975).

Theorem 1.3.6.4.3:

Let X(t) be a submartingale such that


supE(x(t))(a(t)) is replaced by < ) .

In the cases (1.86) and (1.87), the processes T(t) and p. (t) tend to finite limits.

62


Proof: Formula (1.76) contains all the information:

E,(T(t + l)) = T(t) + a(t)

>T(t)

hence a submartingale. Taking expectations of both sides of (1.76) (and noting that E(E t ) = E, (E t ) = E since E is the overall average) we get

E(T(t + l)) = E(T(t)) + E(a(t)).

for all t. Hence

E(T(t))=E(T(t-l)) + E(a(t-l))

= E(T(t-2)) + E(a(t-2)) + E(a(t-l))

= E(T(l)) + f>(a(i))

= l + gE(a(i)). i=i

Hence, condition (1.85) is satisfied if and only if

£E(a(t)) 0 (if p( = 0 the symbol (or more general, the object) is considered as non-existing, hence deleted from the index set {l,...,m}) and to

70


i=l

since we limit ourselves to all actually occurring symbols. It is well-known that H is maximal if and only if all p{ are equal: p, = p 2 = ... = p m = p = — in which case (1.97) m reduces to (1.94) - see e.g. Jones (1979) or Mansuripur (1987) for a reference of this wellknown result. From now on we will not limit ourselves to letters or words but we will treat the probabilities p,,...,p m in a general way being probabilities of occurrence of m objects r = l,...,m.

1.3.7.2 The Principle of Least Effort (PLE) and its relation with the law ofLotka

Besides an average information content, expressed by H in (1.97), we have also an average effort (also called cost or sometimes (in the area of physics) energy) expressed by

E=J>rPr

(1.99)

r=l

where E7 is the effort of using object r. In the example of a text consisting of words, Er is the cost of using the word on rank r, where words are arranged according to increasing length. In this setting it is clear that the E,,...,Em are increasing. Our application is more general, however, but it will be convenient to assume that the E ]; ...,E m are increasing. This is no loss of generality: simply arrange the objects r = l,...,m such that the E,,..., Em are increasing.

Definition 1.3.7.2.1: The Principle of Least Effort (PLE) The principle of least effort requires E in (1.99) to be minimal, subject to the restriction (1.97) (a fixed given value of H ) and the restriction (1.98). Intuitively this means that, within a fixed information content, we seek this situation which requires the least effort.


71

The (PLE) is attributed to Zipf (1949) who promoted (PLE) but Zipf did not present a mathematical formulation as above. In its historical formulation, (PLE) is to be understood as a sociological behavior of persons and groups of persons, see also Petruszewycz (1973).

We have the following proposition.

Proposition 1.3.7.2.2 (Egghe and Lafouge (2004)): (PLE) implies

pr=cp-E'

(1.100)

r = l,...,m, 3 c > 0 , 3 p > l . Proof: Since (PLE) is a constraint extremum problem a necessary (but not always sufficient) condition for (PLE) to be satisfied is given by the method of the multiplicators of Lagrange (see, e.g. Apostol (1957)): consider the function

G = £ E r P r + X , H + £p 7 logp r \ + X2 1 - ^ P r r=l

I

r=l

)

\

r=l

a-101) I

(i.e. minimise (1.99), subject to the constraints (1.97) and (1.98)). The methodology requires = 0 for all r = l,...,m. Hence

| ^ = E r +X 1 (log P r + l ) - ^ 2 = 0

hence (taking log = loge = In but any other base will yield the same result (1.102) below).

pr=e1'

e k'

72


which is of the form (r = 1,... ,m)

pr=cp"E'

where c = e 1 '

(1.102)

> 0 and p = e1' > 0 . Note that ^ , ^ 0 since Xl = 0 implies p — oo and

hence p r = 0 for all r = l,...,m, by (1.102), contradicting (1.98) (and the fact that all probabilities are strictly positive). So we proved (1.100) but only for p > 0. That p > 1 will follow from (PLE) as we will show now. By (1.101), p > 1 will be proved if we can show that the p,,...,p m decrease (since we assumed the E,,...,Em to be positive and to increase). Let p vt 1. Suppose that the p,,...,p m satisfy (PLE), hence are of the form (1.102) and that they do not decrease. Let then i < j , i,je{l,...,m} be such that P; < Pj. Define % to be the elementary permutation of {l,...,m} defined as

(n(l),...,7t(m)) = (l,...,i-l,j,i + l,...,j-l,i,j + l,...,m)

(1.103)

- X > r l n P r = H = - EP,WtaP»(r)

( L104 )

Then we have that

r=l

r=l

and

EPr=1

=

EP* W

r=l

r=l

But

E E 'P«r) = EiP*(.) + EjP^j) + E E r P , W r=l

r=l

( L105 )


73

= E iPl + E JPJ + g E rP 0 , 3 p > 0 .

(1.109)


11

Proof: Using again the method of the multiplicators of Lagrange for the constraint extremum problem (MEP) (see Apostol (1957)) we consider the function

m

I

m

\

(

m

\

G = - g p r l o g p I + X I E - £ ) E r P r +A.2 l - £ ) p r

The requirements

(1.110)

= 0 for all r = 1,.. ,,m now yield (again we take log = In) c>Pr

|^- = -l-lnpr-X,Er-X2=0 "Pr hence

P r =CP^ E '

where c = e~'~Xl > 0 and p = eX| > 0 for all r = l,...,m, hence (1.110).

U

It is not possible now to prove sharper limits for p (as could be done in case of (PLE)). This is because E contains the parameters E,,...,Em while H does not contain any parameters. In fact in the next subsection we will show that the necessary condition for MEP expressed in Proposition 1.3.7.3.2, is also sufficient for (MEP). There, also the sufficiency will be proved for the necessary condition for (PLE), expressed in Proposition 1.3.7.2.2, so that we arrive at the conclusion that (PLE) and (MEP) are not equivalent. However, in the next subsection, the exact relation between (PLE) and (MEP) will be established.

Note that (MEP), via Proposition 1.3.7.3.2, using Er = logr as in Theorem 1,3.7.2.3 leads, as in this Theorem, to a power law but one that is not necessarily decreasing: if 0 < p < 1, then (1.109) increases which is not conform to Zipf s or Lotka's law.

78


For more on (MEP) we refer the reader to Bashkirov and Vityazev (2000) and to Lafouge and Michel (2001) (see also Yablonsky (1980), Haitun (1982c) and Nicolis, Nicolis and Nicolis (1989) for a definition of (MEP) using continuous variables). In Lafouge and Michel (2001) other functions for the parameters Er (other than the logarithmic functions (1.107)) are tried (e.g. the linear one) yielding non-power type functions for p r (or p n ) such as the geometric distribution. As noted above, however, the use of a logarithmic functionality for the E r (r = l,...,m) is the most important one and this leads to power type laws for p r and pn (i.e. the law of Zipf, respectively the law of Lotka).

1.3.7.4 The exact relation between (PLE) and (MEP)

All the results in this subsection are recently proved in Egghe and Lafouge (2004).

In the previous subsection we proved already that (PLE) implies pr = cp~E', 3c> 0, 3p > 1, r = l,...,m and that (MEP) implies p r =cp~ E ', 3c> 0, 3 p > 0 , r = l,...,m . Since (MEP) refers to p > 0 and (PLE) refers to p > 1 we conjecture that, in the determination of the relation between (PLE) and (MEP) another principle is needed. We will define it now and then show that indeed this third optimization principle is the "missing link" in the establishment of the mathematical relation between the principles (PLE) anf (MEP).

Definition 1.3.7.4.1: The Principle of Most Effort (PME) (Egghe and Lafouge (2004)): The principle of most effort requires E in (1.99) to be maximal, subject to the restriction (1.97) (a fixed given value of H) and the restriction (1.98).

This principle (PME) is introduced in Egghe and Lafouge (2003) for the first time. We have the following necessary condition for (PME).

Proposition 1.3.7.4.2 (Egghe and Lafouge (2004)): (PME) implies

P r =cp" E
0, 300, 3p > 0, implies (MEP)

(1.114)

80


(ii) pr=cp-E'

(1.115)

r = l,...,m, 3c > 0 , 3p > 1, implies (PLE) (iii) P r =cp- E
0 , 3 0 < p < 1, implies (PME).

Proof: Given one of the situations (1.114), (1.115) or (1.116), define the following function fr of the variable x r > 0, r = l,...m:

f r (x r ) = x r l n x r - ( l + lnc)x r + (lnp)E r x r

(1.117)

Then we have that

f'r (x r ) = 1 + lnx r - 1 - l n c + (lnp)E r = 0

implies that

_ lnc-(lnp)Er v A —C

xr =cp" E l

r = l,...,m, hence the given functions (1.114), (1.115) or (1.116) respectively. Furthermore

fr"(xr) = — > 0 X


81

for all x r . Hence the given functions (1.114), (1.115) and (1.116) satisfy f^(pr) = 0 and f r ( p r ) > 0 , hence fr has a minimum in xr = pr with p r as in (1.114), (1.115) or (1.116) respectively, r = l,...,m. So for all x r > 0, r = l,...,m:

f,(x r )>f r ( P r )

i.e.

x r l n x r - ( l + lnc)x r +(lnp)E r x r > p r l n p r - ( l + lnc)p r +(lnp)E r p r

(1.118)

Hence

Jx r lnx r -(l + lnc)Jxr+(lnp)jErxI r=l

r=l

T=1

m

m m

>I>rtop r -(l + lnc)£>r + (lnp)£;Erpr r=l

r=l

(I.H9)

r=l

The proof is now split into 4 parts: (I) represents the proof of (i), (II) represents the proof of (ii) for p =* 1, (III) represents the proof of (iii) for p * 1 and (IV) represents the proof of (ii) and (iii) for p = 1.

(I)

Let now (I.I 14) be given: p > 0 and require

r=l

r=l

jE r x r =jE r P r =E r=l

r=l

be constants. Then (I.I 19) implies, for any p > 0:

82


-^x r lnx r 0 ) satisfy the (MEP). This completes the proof of (i). (II)

Let now (1.115) be given and let p ^ 1: so p > 1 and

m

m

r-l

r=l

lllX

-LX

r = "I]Pr lnPr =

H

be constants. Then (1.119) implies, for p > 1 (hence lnp > 0):

jE r x r >J E r P r . r=l

r=l

Hence the p,,...,p m of the form (1.115) ( p > l ) satisfy the (PLE). (III)

Let now (I.I 16) be given and let p ^ 1: so 0 < p < 1 and

r=l

m

r=l

m

~J2Xr

lllX

r=l

r = -£Pr

ln

P, =

H

r=l

be constants. Then (LI 19) implies, for 0 < p < l (hence lnp 0 , 3p > 0

(II)

(PLE) is equivalent with

p r = cp- E '

r = l,...,m, 3 c > 0 , 3p > 1 (III)

(PME) is equivalent with P, = cp~E'

r = l,...,m, 3 c > 0 , 3 0 < p < l The case p = 1 is the only one in the intersection of (PLE) and (PME) and corresponds to the (degenerate) case of the (MEP) where an unconstrained (w.r.t. E) maximum is obtained for H.

Finally, from Theorem 1.3.7.4.4, we obtain the exact relation announced in the heading of Subsection 1.3.7.4.

Theorem 1.3.7.4.5 (Egghe and Lafouge (2004)): The following assertions are equivalent:

(i)

(MEP)

(ii)

(PLE) or (PME).

Note 1.3.7.4.6: As follows from Theorem 1.3.7.4.4 and from the argument developed in the proof of Theorem 1.3.7.2.3, the (MEP) does not always produce a decreasing power law if (1.107) is applied (only (PLE) does). In this sense, the papers Yablonsky (1980), Haitun (1982c) and Nicolis, Nicolis and Nicolis (1989), which use (MEP) (but with continuous variables), do not provide a complete explanation of Zipf s or Lotka's law as a decreasing function.


1.4

85

PRACTICAL EXAMPLES OF LOTKAIAN INFORMETRICS

1.4.1 Important remark

In this section we will not be involved with the statistical fitting of Lotkaian laws (power functions). First of all, because of the non-Gaussian nature of these laws (i.e. the Central Limit Theorem is not valid here due to infinite moments - see e.g. Yablonsky (1980, 1985), Haitun (1982b) or Huber (2001)) it is not possible to perform accurate statistical tests on power laws (of course, as will be explained in the Appendix III, it is possible to make some "optimal" calculations of the values of C and a in formula (1.7), which has practical value). Secondly, and more importantly, a statistical fitting of such laws does not provide evidence for its mathematical acceptance. For the latter, one needs arguments as the ones given in section 1.3.

It is even so that - in statistical fitting - it is very well possible that more than one function can be accepted. The author can refer here to the "old joke" that one provides data points of the form l x , v x ) , say for x = 1,2,...,20 (hence following perfectly the function y = Vx) which can be fitted e.g. by a straight line, hereby "proving" the linear relationship! This is not showing the weakness of statistics but merely the fact that it serves another goal (than the mathematical-probabilistic goal of this book), namely to provide a variety of frameworks in which one can possibly work. Thirdly, even in cases where Lotka's law evidently is not valid (e.g. in cases where f(n) is not overall decreasing but increases and then decreases as in some examples of number of authors per paper - see further in this section) we can use Lotka's law as a first approximation or as a simplification in an otherwise too intricate mathematical theory. This will be done e.g. in the modelling of fractional author scores (Chapter VI) where the proved model is so stable that a simplification as mentioned above does not jeopardise the quality of the final model.

So in this section, we mainly give an overview of areas (also outside informetrics) where power functions for the size-frequency function f are possible or have been studied. In essence, the basic requirements are given in the Axioms 1.3.1.2, together with the scale-free property which then (via Corollary 1.3.2.3) guarantees a decreasing power law. It is the

86


author's opinion that, instead of each time performing statistical tests (on which, as said, there is criticism) one better checks if, in a certain field or application, the scale-free property applies, in which case one has even mathematical evidence of the validity of a power law. This is also a suggestion made in Roberts (1979) where one gives numerous "psychophysical" examples of power laws but where one does not try to fit a power law but merely checks the scale-free property, by experimentation (we come back to this in this section: Subsection 1.4.7).

1.4.2 Lotka's law in the informetrics and linguistics literature

We start this overview by giving some examples in the field of informetrics itself and in its "neighboring" field of linguistics.

The well-known bibliography collected by Hjerppe (1980), covering 80 years of bibliometrics and citation analysis (say in modern terms: informetrics) was analysed in Alvarado (1999) and found to be conform with Lotka's law. As in Lotka (1926), they used straight counts (giving only a credit of 1 to the senior author and nothing to the other authors) to the solution of the multiple author problem. The found exponent a of the power function was around a = 2.3.

In library and information science, there is the study of Sen, bin Taib and bin Hassan (1996), using the LISA (Library and Information Science Abstracts) database in the years 1992 and 1993, resulting in a confirmation of Lotka's law with rather high values of the exponent around a = 3.2. Schorr (1974) collects data in library sciences (in the journals Library Quarterly and College and Research Libraries, 1963-1972) but his method to deny Lotka's law is far from scientific. Yet, in another paper Schorr (1975a), using the same intuitive approach, he accepts Lotka's law in map librarianship (but see the criticism in Coile (1977)).

In Voos (1974), H. Voos studied Lotka's law in information science, using the ISA (Information Science Abstracts) database in the years 1966-1970. Again high exponents around a = 3.5 were found. See also Coile (1975) for comments on these high values.

In Sharada (1993) the field of linguistics was examined using some abstracting journals in this field (1977-1987). Only Lotka's law with exponent a = 2 was examined and found to be acceptable. We will not report on the many studies in linguistics of Zipf s law, which is basic


87

in this field. For this, see Baayen (2001), Herdan (1960), Kanter and Kessler (1995). The same goes for the econometric equivalent of the law of Zipf, being Pareto's law. For this, see the vast econometric literature. Lotka's law ( a = 2) also seems to be confirmed for the use of Chinese words. For this see Rousseau and Zhang (1992). The relativity of these results is clear from the following: in the same publication (Rousseau and Zhang (1992)) Zipf s law (although part of Lotkaian informetrics) is "rejected" for the rank-frequency distribution of Chinese words but "accepted" in Shtrikman (1994). In Rousseau and Rousseau (1993), Lotka's law is confirmed in an example of song texts.

In Newby, Greenberg and Jones (2003) Lotka's law is confirmed for the distribution of "authors" (programmers, open source developers) of open source software.

A recent study, Pulgarin and Gil-Leiva (2004), shows the validity of Lotka's law in the literature on automatic indexing in the period 1956-2000. The found exponent a was 2.75

1.4.3 Lotka's law in networks

It is clear that this is an important example: networks (having their mathematical basis in graph theory) gain increasing interest and importance: intranets, the internet and WWW but also citation networks and collaboration networks (see e.g. Egghe and Rousseau (2002, 2003a)). A network (more generally a graph) consists of points (called vertices or nodes) and connections between points (called edges). Graphs can be unidirectional in case, if (a,b) is an edge (between the points a and b), (b,a) is not an edge; if this restriction is not applicable we call the graph bidirectional. An example of a unidirectional graph is an article-citation network, where the vertices (nodes) are papers and the (one-way) links are citations (although counter-examples (due to "invisible colleges") exist we can, quite logically, assume that if paper a cites paper b, the reverse is not true because of the time difference). An example of a bidirectional graph is the so-called collaboration graph: here vertices (nodes) are authors and there is a link between them if they collaborated, i.e. if they wrote an article (or so) together. Note that this is an example of a weighted graph in which the weight of the edge is equal to the number of documents in which both authors collaborate. Other examples of graphs can be given e.g. the well-known WWW where vertices are web pages and edges the links between them (e.g. via the clickable buttons or hyperlinks).

88


All these networks are examples of so-called "social networks" in which the size-frequency law, being P(k) = the fraction of nodes with k edges (i.e. the degree of connectivity of a node) follows a power law. Their exponents a vary between 2 and 4 (see Bilke and Peterson (2001), Jeong, Tombor, Albert, Ottval and Barabasi (2000), Barabasi, Jeong, Neda, Ravasz, Schubert and Vicsek (2002), Adamic, Lukose, Puniyani and Huberman (2001), Barabasi and Albert (1999) and the excellent recent book Pastor-Satorras and Vespignani (2004)). For reasons explained in Subsection 1.3.2 such networks are called scale-free networks and this property is very important e.g. in the determination of their fractal dimension (i.e. their complexity - see Chapter V). These social networks are in contrast with random networks in which links (edges) between the nodes are given in a random way, given a fixed probability p for a link. These so-called Erdos-Renyi networks (see Erdos and Renyi (1960) or Bollobas (1985)) show an exponential decay for P(k) (see Jeong, Tombor, Albert, Ottval and Barabasi (2000) or Barabasi, Jeong, Neda, Ravasz, Schubert and Vicsek (2002)). The latter result is refined in Krapivsky, Redner and Leyvraz (2000) where one shows the following result for random networks. If the probability to add a new link to an edge that has already k links is dependent on k, say Ak (cf. the SBS-principle of Subsection 1.3.6!) then, if Ak is proportional to k', P(k) is an exponential distribution only if y < 1. For y > 1, nearly all nodes connect to nearly all other nodes and if y = 1 , P(k) is found to be of power type with exponent a variable in the interval ]2,+°°[.

The rank-frequency version of the function P was found to be Zipfian (hence P is Lotkaian) in Adamic and Huberman (2002). There, some consequences of this fact are given. The powertype of the function k —> P(k) implies the large degree of skewness meaning that many nodes have few (say just one or two) connections while a few nodes have many connections. This is judged in Adamic and Huberman (2002) as a two-edged sword as far as resilience of the network is concerned: if a node fails at random it is most likely one with very few connections, hence its failure will not affect the performance of the network overall. However, if one targets a few (or even one) of the high degree nodes, their removal would require rerouting through longer and less optimal paths. If a sufficient number of high degree nodes are removed, the network itself can become fragmented (i.e. disconnected) without a way to communicate from one location to another.


89

In WWW it is reported (Aida, Takahashi and Abe (1998) and references therein) that the number of URLs which are accessed n times (in a certain time period) is a classical Lotka law with exponent a = 2. Note the clear scale-free property of this example. The latter publication shows the need for a standardization of the terminology (cf. also the pleas of Glanzel (1996) and Rousseau (2002a), already mentioned in Section I.I): they do not use the name of Lotka but call its law, together with Zipf s law the "dual Zipfian model". Note that the term "dual" conforms with our dual notion between sources and items and with the duality between the size-frequency function f and the rank-frequency function g, as explained in the same section I.I.

In Kot, Silverman and Berg (2003) one finds that the rank-frequency function, describing the relation between the rank of contributors to a newsgroup and the number of their submissions to that newsgroup, clearly is Zipfian.

In Egghe (2000d), the author gives an overview of regularities in the internet as found in several (electronic) texts. Nielsen (1997) gives evidence that websites, ranked according to their page views per year, follow Zipf s law. Huberman, Pirolli, Pitkow and Lukose (1998) report on the distribution of the number of web pages that a user visits within a given website, to be (approximately) of power law type with exponent around 1.5. One talks in this connection about "the law of surfing" (see also Huberman (2001)).

Adamic and Huberman (2001, 2002) (see also Huberman (2001)) give evidence for Zipf s power law for the rank of websites versus 4 attributes: number of pages, number of users, number of out-links (i.e. hyperlinks given by the sites), number of in-links (i.e. hyperlinks pointing to the sites). The latter (on in-links), in size-frequency version, is also studied in Bornholdt and Ebel (2001): the found Lotka function has an exponent around 2.1. This is confirmed in Albert, Jeong and Barabasi (1999) and Barabasi, Albert and Jeong (2000) where also the size-frequency distribution of the out-links in WWW is studied: here a power law (Lotka function) is found with exponent equalling 2.45. Size-frequency functions of in- and out-links are also studied in Thelwall and Wilkinson (2003) and found to be Lotkaian although no indication of the exponents' values is given. In Faloutsos, Faloutsos and Faloutsos (1999) one studies both size- and rank-frequency functions for out-links and one finds in both cases values of the exponents. Of course, as we will see in Chapter II, one type of exponent implies the other which is not noticed in the mentioned article. This again

90


underlines the need for a concise theory on size- and rank-frequency functions, to be developed from Chapter II onwards. Their values for the Lotka exponent a are around 2.2. In Adamic and Huberman (2002) one finds Zipf s law also in the relation of website ranks versus the number of visitors (via the provider America Online) on December 1, 1997.

Rousseau (1997) gives statistical arguments for the fact that the distribution of hyperlinks between websites follows Lotka's law with exponent in the order of 2.3 (similar to the findings on social networks described above). The same goes for the distribution (occurrence) of domain names (such as .edu, .com, .uk and so on); here an exponent of 1.5 is found. Rousseau calls hyperlinks "sitations", an interesting name referring to "sites" and the more classical "citations". On the latter, we can report on the following results.

Redner (1998) reports on the number of papers with n citations (from ISI data on papers published in 1981 and data of the journal Physical Review D, vols. 11-50) being a power law with exponent a ss3. The far more limited study of Momcilovic and Simeon (1981) reports on a Lotka law with exponent a = 2. The paper of Katz (1999), rightly stressing the importance of a scale-free theory (hence of Lotkaian informetrics) studies the connection between the number of papers that one publishes and the number of citations (recognition) that one receives. Note the ratio scales in this type of problem. Katz finds a power law with exponent around 1.3. Several Zipf-type laws (rank-frequency functions), which are also scalefree, are also mentioned in Katz (1999).

1.4.4 Lotka's law and the number of authors per paper

The classical formulation of Lotka's law is in terms of authors (as sources) and the number of articles (items) they have published. In a dual way (cf. Section I.I), interchanging sources and items, one can consider articles as sources of authors (= items), in other words one can consider the size-frequency function f describing the number (or fraction) of articles with n authors. So, this is a very important topic since it deals with the intricate problem of multiple authorship (see also Chapter VI).

One constatation is very clear: in some cases the number of articles with n authors is decreasing in n = 1,2,3,..., and in other cases there is a short increase in n followed by a long decrease in n. In Gupta, Kumar and Rousseau (1998), 11 of the 12 data sets (in biology) are


91

decreasing; the other one has a small increase until n = 3 and then decreases. In Rousseau (1994c), 7 data sets in information science were examined and all data sets are decreasing in n. In Ajiferuke (1991), 16 data sets were analysed but it is not indicated whether they are decreasing or not. In the topic of information systems, Cunningham and Dillon (1997) gives a data set consisting of 5 journals showing a small increase up to n = 2 from which the decrease starts. Finally, in sciences (physiology), data in Kuch (1978) show an increase up to n = 2 from which the decrease starts.

In general one can conclude that the humanities and social sciences often show a pure decrease, due to the fact that collaboration is lower than e.g. in "hard" sciences, such as physical and life sciences (cf. also the conclusions in Cunningham and Dillon (1997)).

Next there is the issue of which size-frequency function f = f(n) can be applied to the number of articles with n authors. Even in the decreasing case, most authors seem to indicate, based on their statistical fittings, that Lotka's law does not apply. Yet a power law is the simplest function to work with (among the other possible size-frequency functions tested, e.g. forms of Poisson, geometric, Waring, negative binomial,...). Secondly we indicated already the problem of fitting a power law. So, it is, for mathematical purposes, not clear that Lotka's law, as a basic simple expression, should be rejected. To formulate it otherwise: to make a mathematical theory of size-frequency functions and derived properties for IPPs, we want to make as few assumptions as possible, e.g. restricting ourselves (basically) to Axioms 1.3.1.2. But for a mathematical theory we need a workable simple analytical form for the sizefrequency function f. We explained already that a power law is the simplest form (hereby excluding decreasing exponential functions - see Subsection 1.3.3) and hence that the Lotkatype law is the first to be used in models although real data sometimes seem to deviate from it. This is the more true if the distribution of the number of authors per article is not decreasing but shows a fast increase (up to a small value of n = the number of authors per article, usually n = 2 o r n = 3 - see the examples above) and then a decrease from this n onwards. Lotka's law can be taken here as a first approximation of this regularity (a bit comparable with lognormal ageing curves - see Mattricciani (1991) or Egghe and Rao (1992a), being approximated by decreasing exponential ageing (obsolescence) functions).

In conclusion: most of the co-authorship data show a decreasing frequency function f; the others show a quick small increase (usually up to number of authors n = 2 or 3) after which

92


the decrease is clear. In both cases we will use - as a first approximation - the power law function as an analytical description. This will enable us to derive mathematical results as e.g. in the case of the fractional frequency distribution; i.e. the size-frequency distribution if authorship in multiple-authored papers is credited fractionally (see Chapter VI), where both size-frequency functions: the number of papers per author and the number of authors per paper, are used and taken as decreasing power laws. The obtained results are then assumed (due to an assumed robustness of the model) also to be valid for the general case of Axioms 1.3.1.2, including - of course - other size-frequency functions f. Note (again as in Subsection 1.3.1) that this approach is in line with the strong request of Griffith (1988) for simple explanations.

In Gupta and Karisiddippa (1999) a variant of collaboration is studied, namely the number of collaborators (co-authors) per author. One found a very classical power law with exponent a = 2.1 for the number of authors with a certain average number of collaborators. This variant was also studied in Qin (1995) and further studied in Chapter III.

1.4.5 Time dependence and Lotka's law

As in any informetrics (or even other types of -metrics) problem, time is an important issue. Adding a time dimension to the study is transforming the problem (and its solution) from a static to a dynamic one. We encountered already an example in the study of SBS (Subsection 1.3.6), but also Naranan's model (Subsection 1.3.4) deals with time as a (main) factor in explaining power laws. In all these models a source, after creating a first item, can, from time to time, produce another item. Hence, at any time t, in our IPP, we have a mixture of "living" sources, some very young, others older. Some authors as Huber, Wagner-Dobler and Berg (references are given in the sequel) (see also Huberman (2001)) have posed some important and interesting questions around this mixture. Expressed in terms of authors (as sources) and their productivity (i.e. publications as items) they rightly state that (in our terminology) in general IPPs there are authors (or inventors) who just start their careers and older inventors, where sometimes a whole career is covered. The article Wagner-Dobler and Berg (1995) agrees with the validity of Lotka's law for such general ("mixed") IPPs (they present many examples of authors in mathematical logic, where their productivity (number of publications) follows Lotka's law (with exponents in the range 1.4-1.7)). On the other hand they pose the interesting question of how is the size-frequency function changed if we restrict our IPP to


93

authors with a fixed number of years of activity (in their examples: 6 respectively 10 years). Then their data show a deviation from Lotka's law, especially for the production quantities n = 2,3,4,5 (n=l was not taken in consideration) which are "more equal" than in Lotka's case which came out in a "mixed" IPP of career duration but where a time period of publications was also fixed at 6 respectively 10 years. This is remarkable and needs a mathematical explanation, which is not available at this moment, at least to the best of the author's knowledge.

In the same direction goes the work of Huber: Huber (1998c, 1999, 2001, 2002) and the articles Huber and Wagner-Dobler (2001a,b). The intention is to "breakdown" an IPP, hereby only analyzing "homogene" parts e.g. of equal career duration and then studying also the distribution of career duration. The methodology applied in Huber (1998c, 1999, 2001, 2002) and Huber and Wagner-Dobler (2001a,b) is statistical and by simulation and the following ingredients are used: most authors tend to produce at a constant rate over their careers, rankfrequency distributions of career duration are exponential, time pattern of outputs within a career is random (hence Poisson distributed) and rank-frequency distributions of pure productivity i.e. rate of publication, i.e. number of publications in a unit time interval (e.g. a year) also is exponential. The double use of the exponential rank-frequency distribution is certainly in view of Subsection 1.3.3 - not very clear to the present author, but the division of a mixed IPP of author (or more general any source) production as above is certainly interesting and needs to be studied from a mathematical modelling point of view.

Indeed, while there is nothing wrong with the study of general "mixed" IPPs, i.e. where sources of any "lifetime" are featuring, such a kind of study is (as Wagner-Dobler and Berg (1995) state) a description of the state of a scientific discipline and not of a distribution of psychological characteristics of individuals. We can add to this that one type of study can add new findings to the other type of study and hence leads to a better understanding of informetrics as a whole.

We can close this subsection by concluding that, most probably, Lotkaian informetrics describes the totality of a subject in which sources of different ages can occur, as e.g. also expressed by SBS orNaranan's model (Subsections 1.3.6 and 1.3.4 respectively).

94


1.4.6 Miscellaneous examples of Lotkaian informetrics

Of course there is the original article Lotka (1926) itself which is also discussed in De Solla Price (1963). De Solla Price tries to understand the universally applicability of Lotka's law but is not very successful. It is in these discussions that De Solla Price is formulating (implicitely, see p. 46) his well-known Price law of concentration. This will be discussed in a rigorous way in Chapter IV. There we will shed light on the "feeling" of De Solla Price (p. 49-50) that Lotka's law is related with the so-called law of Weber-Fechner (from psychology) stating that sensation is proportional to the logarithm of the stimulus (as was discussed in more detail in Subsection 1.3.7.1).

Lotka's law has been confirmed in physics (1800-1900) in Wagner-Dobler and Berg (1999), both for the number of authors versus number of papers (exponent a = 1.9) as for the number of authors versus the number of areas in which they published (exponent a = 2.1). So we can say that the classical Lotka function f (n) = — was found. In semiconductor literature (1978n 1997) also a = 2 is found but only when using truncated data, see Tsay, Jou and Ma (2000). In Indian physics, Gupta, Sharma and Kumar (1998) report on weak applicability of Lotka's function but the results are not checkable since the data are not available. In Gupta (1989) (not the same Gupta as in the previous reference) Lotka's law was confirmed in biochemical literature of Nigeria (1970-1984), where exponents in the range a = 2.0-2.6 are found. A general criticism is needed here (as well as for the previous paper and several other ones): after checking Lotka's law, they continue to check derived properties such as the 80/20 rule (see Chapter IV). There is no reason for this since inequality measures (such as the (generalized) 80/20 rule) are a mathematical consequence of laws like the one of Lotka (see Chapter IV) and hence should not be checked in addition to checks for Lotka's power law.

In Gupta and Kumar (1998), Lotka's law was confirmed in 11 core journals in theoretical population genetics (1971-1980) with exponents in the range a = 1.6-2.4 (so, roughly around a = 2). The same was done in Gupta and Karisiddippa (1996) for theoretical population genetics in the period 1900-1980 with exponent values close to a = 2 (again the 80/20 rule was, unnecessarily, checked).


95

In drug literature there is the paper Windsor (1975) reporting on the applicability of Lotka's classical law ( a = 2) but one did not check for better fitting a s . In legal medecine, Schorr (1975b) rejects Lotka's law but on non-scientific grounds (and only checking a = 2). In the Spanish medical literature Lotka's law, with exponent a close to 2, is confirmed in Terrada andNavarro (1977).

In computer science there is the often cited paper Radhakrishnan and Kernizan (1979). Papers in the period 1968-1972 in the journals Communications of the Association for Computing Machinery (ACM) and the Journal of the ACM where checked for Lotka's law and accepted for a around 3 (per journal). But they recognise the limitation of their study and suggest further experimentation. The high value of a = 3 was explained in Subramanyam (1979) by the fact that Radhakrishnan and Kernizan used total counting of authorship (see also Chapter VI for the treatment of multiple authorship) with the consequence of a high number of authors with 1 publication, pushing up the value of a (this is correct as will follow from Chapter IV); they indicate that, if only senior authors are counted (as did Lotka in his historic paper Lotka (1926)), values of a more close to 2 are found. The bibliography of microcomputer software of Nieuwenhuysen (1988) was checked for Lotka's law and found to be acceptable in Rousseau (1990b) with a around 2.2. Bogaert, Rousseau and Van Hecke (2000) even found Lotka's law in percolation models (where, in a 2-dimensional square lattice with mxm cells there is a probability p for a cell to be occupied). Then one can check the sizes of the soformed clusters of occupied cells and the corresponding size-frequency distribution. In Bogaert, Rousseau and Van Hecke (2000) one confirms Lotka's law with an exponent (evidently) dependent on p. They (experimentally) found the remarkable relation (a power law itself!)

«=

6 18

-

l,

(100p)0311

(1.120)

showing that only Lotka's exponent a is enough to model this phenomenon.

A special area of scientific information is given by the size-frequency function of patents. That it is of Lotkaian type was shown (experimentally) in Huber (1998a) via the Pareto distribution, which belongs to Lotkaian informetrics, as mentioned in Subsection 1.2.2 and as

96


will be proved in Chapter II (via Zipf s law, which is mathematically the same as Pareto's law, see Subsection 1.2.2).

The publication activity of an institute is studied in Vinkler (1990) and confirmed to be Lotkaian (although no indication of its exponent is given). Of course here, the activity of an institute is measured according to the distribution of papers of the researchers in this institute. In this context, the paper Coleman (1992) is interesting. There, author productivity (authors from 101 laboratories) is compared with lab productivity. In both cases Lotka's law was acceptable but with different exponents: for authors the exponent was around 1.9 while for labs it was only around 1.1. A rationale (at least for the difference of both IPPs) will be given in Chapter III where positive reinforcement will be studied in the connection with threedimensional informetrics. Another study was undertaken in Kumar, Sharma and Garg (1998) where Lotka's law (with a = 2) was confirmed for industrial firms but denied for institutional productivity in engineering sciences.

The rank- and size-frequency distributions of misprints in references to a paper (that had acquired 4,300 citations) are found in Simkin and Roychowdhury (2002) to be Zipfian/Lotkaian respectively, with Lotka's exponent a ~ 2 . The fact that many mistakes are repeated frequently is explained in Simkin and Roychowdhury (2002) by the fact that many references are given by reproducing earlier references (hence "copying" the same mistakes) without reading the papers themselves. Based on a model given in Simkin and Roychowdhury (2002) the authors conclude that only about 20% of citations have actually been read by the citers!

For firm sizes, Zipf s law with exponent pS = 1 (corresponding with Lotka's a = 2 - see Chapter II) was found in Axtell (2001) while the same was done for the distribution of city populations in Marsili and Zhang (1998) (but in a more theoretical way and allowing for more general exponents) and in Ioannides and Overman (2003). The distribution of city sizes even goes back to Zipf (1949) with exponent p = 1 (see also Batty (2003)).

Lotka's law with exponent around 2 is found for the number of hits by artists, see Cook (1989). Journal articles on the American Revolution were analyzed in Pao (1982a) and found to be of Lotka type with exponent around 2.6.


97

In Lemoine (1992), Lotka's law (for a in the range 1.5-2) was found for the productivity of Venezuelan scientists. Distinction was made between male and female scientists and between articles in national or international journals. The found lower productivity of women versus men (except in national journals) is interpreted in terms of different Lotka exponents (higher exponents are found for female production). Note that higher a s only indicate a higher inequality among women scientists (as will be proved in Chapter IV) and not a lower productivity, as argued in Lemoine (1992). A similar study (on the difference between male and female productivity) was executed in Gupta, Kumar and Aggarwal (1999), this time in an Indian research institute. Here, no difference between the productivity between male and female scientists is found (reflecting the high degree of emancipation of women in the Indian scientific community, which is also apparent when one visits scientific institutes in India). Lotka's law was found to deviate from the different data sets.

In Nath and Jackson (1991), Lotka's law was examined for management information systems articles in the period 1975-1987 and a good "fit" with exponent a around 2.7 was concluded.

Potter (1981) reports on monograph productivity based on data of Me Callum and Godwin (unpublished) (Library of Congress Study on author headings on MARC tapes) and of himself in Potter (1980) on personal authors in the University of Illinois library catalogue. Here all authors are counted (Potter explains why Lotka only counted senior authors - see Subsection 1.2.1). Only the exponent a = 2 was checked and reasonable fits were found. Potter (1981) also reviews work of Leavens on papers in the journal Econometrica (1933-1952). Again only the exponent a = 2 for Lotka's law is considered and the latter was confirmed.

The review article White and Me Cain (1989) discusses work of Nicholls (1988), where Lotka's law is found in the humanities and natural sciences ( a = 2.5 on average), and in the social sciences ( a = 3.0 on average). Also here one then checks De Solla Price's law on the unequal division of author productivity (see further) but this is unnecessary since, in Chapter IV, we will see that the exact form of De Solla Price's law can be derived from the one of Lotka; hence separate tests are not necessary. Lotka's law is confirmed (for a = 2) in Murphy (1973) (but denied for this value in Coile (1977), where, however, no further tests using other exponential values are presented) in the humanities (in Technology and Culture, 1960-1969).

98


Let us also refer to Pao (1986) for a review of many of the above mentioned applications but where the Pao testing procedure (see the Appendix III) is applied.

1.4.7 Observations of the scale-free property of the size-frequency function f

If we want to establish the size-frequency function of an IPP it is interesting to consider the data on a double logarithmic scale (for those who work manually: on double-logarithmic paper) and check for linearity, i.e. if the data follow a straight line. The reader might argue that this is exactly what most tests do and that we criticized this methodology - see Subsection 1.2.1 (formula (1.8)) or the Appendix III. The criticism is still there on the fitting of Q

the right power law f (j) = —, i.e. to determine the best C and a . But to decide that we have a power law it is one of the best ways to do. In terms of the size-frequency function f, what we do is to transform f into

F(x) = log,f(a x )

(1.121)

where a usually is taken to be 10 or e (but any a > 1 can be used).

An example of this procedure is shown in Roberts (1979) where one reports on work in Stevens (1960) where tests were done for several possible scale-free (hence power) functions of the type: express (by putting force in a handgrip - the ordinate) the value of a given stimulus (the abscissa). Examples of stimuli were: electric shock, warmth, lifted weights, pressure on palm, cold, vibration, white noise, tones, white light. The results are shown in Fig. 1.4.1.

Note that in these examples no fitting of the power law is done. In the same way one finds examples in Katz (1999) on recognition (citations) versus publishing size or on publishing size versus rank of publishing size (Zipf type functions, which are also scale-free) or recognition (citations) versus recognition rank or impact versus impact rank. We also refer to Brown and West (2000), a book that is completely devoted to scaling and power laws in biology.


99

We encourage the execution of more of these type of studies leading to exact conclusions as to whether or not we have a power function, i.e. Lotkaian informetrics.

Fig. 1.4.1

Testing for scale-free properties (i.e. ratio scales).

Reprinted from Roberts (1979), Fig. 4.4, p. 184. Reproduced with permission from Cambridge University Press.


II BASIC THEORY OF LOTKAIAN INFORMETRICS

II.l GENERAL INFORMETRICS THEORY Before developing the Lotkaian informetrics theory we will describe, formally, the functions that are needed in general informetrics - e.g. a formal description of the functions introduced in Section I.I and we will prove their properties. The basis of this general theory is duality between sources and items, i.e. their interchangeability or the possibility of replacing "produces" by "is produced by". Exact definitions will be given, followed by a heuristic interpretation. We hope, this way, to give also an intuitively clear description of the necessary formalism which, however, limits itself to aspects of real analysis (mainly derivatives and integrals). The reader who wants an update on these techniques is referred to the vast literature on real analysis, e.g. Apostol (1957) or Protter and Morrey (1977). That we use derivatives and integrals is evident from the arguments given in the first chapter: the simplicity of the functional formalism and its calculations (as compared to the evaluation of sums) and also the simplicity of the relations between the different functions and their parameters and further, its applicability to the practical cases of generalized bibliographies since, the larger the datasets (as they are more and more common because of literature growth and the automation of its collection), the better they are represented by continuous models (see also the review article Wilson (1999)). In addition, they allow for fractional counting (rational, non-entire numbers) as will become clear in Chapter VI.

II.l.l Generalized bibliographies: Information Production Processes (IPPs)

An information production process (IPP) is a triple of the form (S,I,V) where S = [0,T] is the set of sources, I = [0,A] is the set of items and where

102


V:SÎ

(II.l)

is a strictly increasing differentiable function such that V(0) = 0 and V(T) = A.

Intuition; S and I stand for the discrete sets of sources {1,2,...,T} and of items {1,2,...,A} respectively, hence the interval lengths (T respectively A) stand for the total number of sources, respectively items. The function V represents the cumulative number of items in the least productive sources. Since we will identify S with the source rankings (in decreasing order of number of items), V(r) (for re[0,T]) denotes the cumulative number of items in the sources belonging to the interval [T — r,T] (i.e. the sources in the interval of length r, containing the least productive sources).

We next introduce the notion of duality. Let (S,I,V) = ([0,T], [0,A], V) be an IPP as defined above. The dual IPP of the IPP (S,I,V) is defined to be the IPP

(I,S,U) = ([O,A],[O,T],U),

(II.2)

U(i) = T - V - ' ( A - i )

(II.3)

where U : I —> S is the function

for all ie[0,A]. Note that V"1 (the inverse of V) exists since V is strictly increasing.

Note also that it follows from (II.3) that i = V(r) « T - r = U(A - i).

Intuition: When replacing the words "source" and "item" in the IPP (S,I,V) we obtain the IPP (I,S,U). Now, for each ie[0,A], U(i) denotes the cumulative number of sources that produce the items belonging to the interval [0,i], for each ie[0,A].

Note that, since V is differentiable and strictly increasing, the same is true for U. Denote, for every iel and reS

Basic theory ofLotkaian informetrics

103

a(i) = U'(i)

(II.4)

p(r) = V'(r)

(II.5)

We will (with an acceptable abuse of notation) denote

p(i) = V'(r)

if and only if i = V(r). In other words

p(i) = V'(v- 1 (i))

(II.6)

for all iel.

Intuition: For each reS, p (r) is the density function of the items in the source T - r and, for each iel, a (i) is the density function of the sources in the item i.

We have the following result:

Proposition H.l.1.1:

C(I)=

for every iel.

Proof: By (II.3)

K^)

(IL7)

104

Power laws in the information production process: Lotkaian informetrics 1

"P(A-0 ' by (II.6).

D

Intuition: Because of (II.7), for every iel, p (i) is the density function of the items in the point A-i.

Our intuitive explanation that S = [0,T] is ordered in decreasing order of number of items is defined exactly by requiring p to be increasing (hence, by (II.7), also a increases). We furthermore suppose p (hence also a ) to be differentiate. Finally (although this is not necessary) we will always suppose p (0) = 1 (the lowest density of items in a source, i.e. the density in source T). Note that (by (II.5) and (II.6)) p (A) = p (T) is the highest density of items in a source (since V(T) = A). We denote this by pm .

II.1.2 General informetric functions in an IPP

We are now in a position to introduce, formally, the four informetric functions, introduced heuristically in sections I.I and 1.2. These basic functions are:

(i)

c itself

(ii)

g defined as g(r) = p ( T - r )

(II.8)

G(r)= f g ( r ' ) d r '

(II.9)

for all re[0,T] (iii)

G defined as

for all re[0,T] (iv)

f is defined as

m

~WUij

R + such that

poo

^ =u < ^

T

'—

(11.23)

I f*0>J

We have, necessarily, that f* = Df with D > 0 a constant.

Proof: (i) ~ (ii)

By (11.22), (11.20) and (11.21), we must show that (taking f = f*)

rjf(j)dj _f f(j)dj

XW /"fojdj

which follows from the Lemma II. 1.3.2 following this theorem.

(ii) => (i)

Consider the function (for x > 1)

f jf*(i)dj cp : x - ^ -

(11.24)

112

Power laws in the information production process: Lotkaian informetrics Using several results of mathematical analysis (see Apostol (1957) or Protter and Morrey (1977)), we have that cp is a continuous function, hence attains all values between its minimum and maximum value, being (since f* > 0) respectively (using de l'Hospital'srule)

limm(x) = lim . . . V

*-i

'

*-i f

=1

x)

and

(since
1, we hence have the existence of a finite number, denoted pm > 1 such that

^ = ^ = dyP——

T

(H-25)

rnj^

Note that pm only depends on \i. From (11.25) it follows that, since f* > 0, there exists a constant D > 0 such that

f " j f ( j ) d j = DA

£f

(j)dj = DT .

Hence, the function f = — satisfies (11.20) and (11.21), hence (i).


113

Lemma II. 1.3.2: For any integrable positive function f we have that 1 < x < y < oo implies that

fXuf(u)du rYvf(v)dv ' <J'

J

J ] f(u)du

J f(v)dv

(H.26)

Proof: It suffices to show that

(j|Xuf(u)du)(^yf(v)dv) T > 0 as (given beforehand) the total number of items, respectively sources in an IPP. So, now on to the theory of Lotkaian informetrics.

114


II.2 THEORY OF LOTKAIAN INFORMETRICS With the preparations of Section II. 1 we can now proceed with Lotkaian informetrics theory. This is, very simply, the theory that can be developed with a size-frequency function f (see Section II. 1) of the form

f:[l,P m ]->R +

j-f(j) = ^

,

(H.27)

where C and a are positive constants, hence a decreasing power function in the density j .

It is logical to start with the existence theory of such functions, given an IPP with A items and T sources. We will base ourselves on the general existence Theorem II. 1.3.1 for sizefrequency functions.

II.2.1 Lotkaian function existence theory

We will first give the easy theory when pm = oo (recall that pm = p (A) = p (T), the maximal item per source density), being a limiting case of the case pm < oo (the realistic case) which we will study in the next subsection. Most of the results go back to Egghe (2004b).

II.2.1.1 The case p m = oo

Let A > T > 0 be given and let f be of the form (11.27). We have the following result.

Proposition H.2.1.1.1 (Egghe (2004b)): The power function f as in (11.27) satisfies (11.20) and (11.21) for p m = oo and A < oo (hence T < oo) if the following relations hold:

9 A —T

cc = ^ — -

A-T

(11.28); V

Basic theory ofLotkaian informetrics AT

C = -^— A-T

115

(11.29)

which implies that a > 2, if A < oo.

Proof: (11.20) and (11.21) yield for pm = oo:

poo C

r-2_dj=A If a < 2 the last relation cannot be satisfied unless A = oo. If a > 2 we have the conditions

^ =T a-1

^ a-2

=A

from which (11.28) and (11.29) follow.

Note that (11.28) and (11.29) can also be read as

a =^

for ^ " Z i

H-l

(

C= -^- , H-l

cn.30)

a-2

(11.31)

where \x = — is the average production. Note that a only depends on \x while C depends on both A and fj. (hence on both T and A).

116


Examples II.2.1.1.2:

1.

A=l,000 , T=500. Then n=2, a =3 and C=l,000 so that the function f(j) = - ^ r satisfies.

2.

A=10,000,T=l,000. Then n=10 and a = — sa2.11 , C = ^ ^ » 9 9 function f ( j ) = ' 2 U

3.

1,111 so that the

satisfies.

99 100 000 A=l 00,000, T=2,000. Then u =50 and oc = — PS2.02 , C = ' ^2,041 so that 49 49 the function f (j) =

] 202 (a "classical" Lotka law with a ~ 2) satisfies.

II. 2.1.2 The general case pm < oo

From Theorem II. 1.3.1 we obtain the following existence theorem for Lotkaian functions, given A > T > 0.

Theorem H.2.1.2.1 (Egghe (2004b)): Let A > T > 0 be given. Let a > 1.

(i)

If a < 2 then there always exists a number p m > 1 and C > 0 such that

satisfies (11.20) and (11.21).

(ii)

If a > 2 then the conclusion (i) is valid if and only if

Basic theory ofLotkaian informetrics 117

^ = M1.6667 = — - . a-2

P

C So, according to the Theorem II.2.1.2.1, now there is no x = pm and function f = —^j that yields A and T. Equation (11.37) now has no solution in x = pm > 1.

The existence theory developed above can also be studied in Egghe (2004b). Note II.2.1.2.4: It follows from (11.23) (or, more simply from 11.14) that it is necessary for the existence of a size-frequency function f, given A > T > 0 and a > 1 that

A<J[°°jf(j)dj .

The right hand side of the above inequality is oo if a < 2 and hence is always satisfied. If a > 2 then the above inequality is

A < ^ a-2 or


a < —+ 2 . A

121

(11.39)

This inequality was already noted in Egghe (1989, 1990a), see also Egghe and Rousseau (1990a).

From the results in this subsection it is also clear that the value of the exponent a = 2 is not only a "classical" value but also a "turning" point in the existence of the size-frequency function f. Throughout this book we will find more properties, implied by the Lotka function f, that have a = 2 as a turning point.

Note II.2.1.2.5: The results in Subsection II.2.1 could be considered as a mathematical "fitting" method for power functions, given A > T > 0, as opposed to statistical fitting (cf. the Appendix III on statistical fitting of a power law where formula (11.28) is re-used to quickly determine a , given a set of practical data).

We now turn our attention to the relations between the functions f, g, G and a as defined in Subsection II. 1.2.

II.2.2 The informetric functions that are equivalent with a Lotkaian size-frequency function f

We will consider the power functions

{

(i) = f

(H.40)

je[l,p m ], C > 0 and a > 0 and study the types of functions for g, G and a that are equivalent with f. We start with the "classical" case a = 2. We have the following theorem: Theorem H.2.2.1 (Egghe (1985,1989,1990a)) n : The following assertions are equivalent: (*) Preliminary, approximative results already appeared in Yablonsky (1980) and Bookstein (1984).

122


(0 f(j) = §

(IL41>

, je[l, p m ], C > 0 a constant: Lotka's function with exponent a = 2.

(ii)

8

«=T4

(IL42)

, re[0,T], E, F > 0 constants: Mandelbrot's function with exponent P = 1.

(iii) G(r) = aln(l + br)

(11.43)

, re[0,T], a, b > 0 constants : Leimkuhler's function.

(iv) a(i) = M.Kj

(11.44)

, ie[0,A], M > 0, K > 1, M, K : constants: Bradford's function.

We, furthermore, have the following relations between the diverse parameters:

E = Pm

(11.45)

F = £a

(11.46)

a= - = C

(11.47)


b= F= -^ C

(11.48)

M=— ab

(11.49)

K = e*

(11.50)

Proof: (iWii)

123

=» We use formula (11.13), yielding

r(j) = g-'(j) = j ; P " ^ d j '

j Pm

Putting j = g(r) we find

implying (11.42) with

E = Pm and

F=P- . C 0 and K > 1 since a,b > 0).

=^p(r-(.4--^f. Hence

a(i) = U'(i) = ( p - - 2 z E L i ) ^ .

«= Since U' = a and since U(0) = 0 we have

U(i) = / o 'a(i')di'

u (i )^|(p---i^) 3 -Pr

Going to the inverse U~' we have

132


whence

G(r) = U~>(r)

2-«

Pm

r

C )

'

showing all equivalencies and all relations between the parameters.

In

Theorems

II.2.2.1

and

II.2.2.3

we

could

have

sufficed

by

only

proving

(i) => (ii) => (iii) => (iv) =^ (i) . However the proof of (iv) => (i) requires implicitely all calculations

involved

in

proving

(iv) => (iii) => (ii) => (i).

Therefore

we

proved

(i) <S> (ii) (iii) <s> (iv), hereby giving also direct proofs of equivalencies.

In Rousseau (1990a) some extensions of the above theorem are proved where other intervals than [l,p m ], [0,T],... are considered, trying to better conform with the discrete case but it is not clear that the results are useful in this case. A similar attempt is undertaken in Bookstein (1990a). Also the discrete index approaches of Chen and Leimkuhler (1986) and Hubert (1978) are not capable of establishing a far reaching informetric theory based on relations between the laws of Lotka, Bradford and Zipf. The early paper of Naranan (1971) has the merit to study rather rigorously the relation between the classical formulation of Bradford's law (i.e. in terms of groups) and functions of the type G.

Comments on the results of Theorems II.2.2.1 or II.2.2.3 can be found in Griffith (1988) and Wilson (1999).

Note H.2.2.4: From Theorem II.2.2.3 we see that, if a > 1 that


133

with p > 0 and F > 0 (also E > 0 ). If a < 1 we have that

g ( r ) = E ( l + Fr)"p

(11.60)

with E > 0, F < 0 and -(3 > 0 (i.e. P < 0).

Although both cases could be proved in one time, regularities (11.59) and (11.60) are different: (11.59) represents the known Mandelbrot function; (11.60) was not included in the classical calculations and are due to Egghe (1985, 1989, 1990a).

Note H.2.2.5: From (11.58) it follows that (hence in the case a > 0 , a * l ,

P(i) = (l + i

^

(11.61)

Proof:

by Proposition II.2.1. Hence

i

i-\

( i a

J. 2 - a

.2-a"!^

P(.) = [p--A—+ i — j 1

M

f

.2-a^

p(i) = [l + . - ^ - j since (11.35) implies that

o*2)

134


- ( 2 - a ) = p2-"-l.

D

Note also that, for all ie[0,A]

1+ i— - > 0

(11.62)

(as it should). This is proved as follows: If a < 2 then (11.62) is trivial. If a > 2 then

r , .2-a , . 2-a . mf l + i =1 + A >0 ie[0,A]

C

C

since (11.39) is valid for a > 2 . So

for all ie[0,A].

D

Exercise H.2.2.6: Show that, in case we have (11.52) in Theorem II.2.2.3 with p m = oo , that

i

pa-l

g(r)=

j_ (r(a-l))-

i f a > 0 , a ^ l , a ^ 2 . From Theorem II.2.2.1 it follows that this result is also valid if a —2 (i.e. g(r) = —) (hint: calculate lim g(r), with g(r) as in (11.53) or reproduce the proof of (i) <s> (ii) in Theorem II.2.2.3). Similar derivations can be made for G and a . As in Theorems II.2.2.1 and II.2.2.3, all functions obtained here are equivalent.


135

We have now characterized the power size-frequency functions f(j) = — for all oc>0, a ^ 1. We will now complete the characterization with a surprising result for a = 1 and we will also present the case a = 0 for the sake of completeness. As is natural (see also Axiom 1.3.1.2 (ii)) for size-frequency functions fin informetrics, we will not consider increasing functions, hence we will not treat the cases a < 0 (although there might be a reason to do so which is not clear at the moment - the above developed techniques can - of course - then be applied).

For the case a = 1, we have the following (remarkable - for reasons to be explained further on) result.

Theorem H.2.2.7 (Egghe and Rousseau (2003b)): The following assertions are equivalent:

(i) f(j) = -

(11.63)

j € [l,p m ], C > 0 a constant: Lotka's power law with exponent a = 1.

(ii) g(r) = Hkr

(11.64)

i

re[0,T],

H = pm>l,

0 0 , 0 < k < 1, N, k: constants. -Ink

(11.65)

136


(iv) cj(i) = — - — W P + Qi

i e [0, A], P = -Nlnk > 0, Q = Ink < 0 : constants.

Proof: (i)(ii)

=» From (11.13) we find

r(j) = g-1(j) = j : P ™ ^ J '

= Cln%] So

p(T-r) = g(r) = j = p m e " c = H k .

1

z

with H = p m > 1, k = e~ e ]0,l[. Hence (11.64) is proved. «- Conversely (11.64) implies g'(r) = H(lnk)k r

r'(i) - h ( l ) Hence, by (11.10) we have

s (g

(j)j

(11.66)

Basic theory ofLotkaian informetrics 137

\_\ Ink j

hence (11.63) with C =

(as found above). Ink

G(r) = / o r g(r')dr'

hence (11.65) with N = — — > 0. — Ink

g(r) = G ' ( r ) = - N ( l n k ) k r

hence we refind (11.64).

(iii) Since

G(r) = U - ( r ) = N ( l - k ' )

((11.18) and (11.65)) we have

U(i) U = — ln(l-i| Ink { Nj

whence

138

Power laws in the information production process: Lotkaian informetrics 1 CT(i)

= U'(i) = - S i p . ~N

, hence (11.66) with P = — Nlnk > 0 and Q = Ink < 0 (note that, because of this, a increases, as it should). 0 and K = — > 1 (again replacing r k byi).

In words: upon adding a translation r —> T — r in the Mandelbrot function, this function and the Bradford function are interchanged when going from a = 1 to a = 2 or vice-versa, a

144


remarkable constatation ! Note that this result could only be revealed by defining the Bradford formalism in a continuous setting, hereby defining a Bradford function which is not defined in the historical formulation (Bradford (1934)).

It is furthermore remarkable that Lotka's power law (for a = 1) for f includes an exponential law for g, see formula (11.64) and not only Mandelbrot-type power laws (of 1 + Fr), see formula (11.53). In this sense, size-frequency functions (of power law type) comprise a wide, heterogeneous, class of rank-frequency functions.

This finishes the "classical" informetrics theory around the well-known functionalities of the type of Lotka, Mandelbrot, Leimkuhler and Bradford. In the next section we will investigate duality properties of these functions and we will make the constatation that there is an informetric function missing: the dual size-frequency function. We will introduce it here for the first time and derive its properties. We will also indicate possible applications.

II.3 EXTENSION

OF

THE

GENERAL

INFORMETRICS

THEORY: THE DUAL SIZE-FREQUENCY FUNCTION h From the previous sections (mainly Section II. 1) we can conclude that duality can be described as: interchanging the terms "item" and "source" and reversing the order.

We repeat the definition of U and V (Section II. 1):

U(i)

=

cumulative number of sources that produce the items belonging to the interval [0,i], for each ie[0,A],

V(r)

=

cumulative number of items in the sources belonging to the interval [T - r,T], for each r e [ 0 , T ] .

This makes a(i) = U'(i) and p(r) = V'(r) dual functions: both increasing and describing the same functionalities: density of sources in an item (for a) resp. density of items in a source (for p), which is also expressed by their relation (II.7). The functions g(r) = p(T —r) and


145

G(r) — I g(r')dr' are simple transformations of p . This reveals the duality of three of the O 0

four functions, defined in Subsection II.1.2: a , g, G, leaving out our basic size-frequency function f. Since f is defined as (11.10) g' = — { drj

f(S

=-Jt?o))=- (8 " 1)1())

j € [l,Pm] and by the above remark, it is clear that another size-frequency function, now based on a , can be defined. Because p(r) = g (T — r), r £ [0, T] and invoking the duality between a and p , we hence formally define the dual size-frequency function h as a' = —

I h(s)=

,,

1

dij

,=(a-')'(s)

(11.76)

a'{a (s))

for s = a(i) e [a(0),l], 1 = G ( A ) (since we took p(0) = 1 and because of (II.7)).

Intuition:

f(j)

=

density of sources in the item density (in a source) j 6 [l,pm]

h(s)

=

density of items in the source density (in an item) s 6 [a (0), l]

and hence are dual functions.

Indeed, for f this has been explained in Subsection II. 1.2, using (11.11). A similar argument yields the interpretation for h: since a(i) is the source density in i € [0, A] (by (II.4)) we have that a"' (s) is the cumulative number of items with source density smaller than or equal to s (since CT increases). Hence h(s) = fa~' )'(s) (by (11.76)) is the density of items in the source

146


density s.

In the discrete setting (which is modelled by the continuous functions above) f(j) is the number of sources with j items and h(s) is the number of items with s (< l) sources, i.e. the number of items produced by sources which have - > 1 items, using the fractional counting s system (see also Chapter VI for a thourough description of this system). Hence it is clear that the fractional counting system is the natural one if one studies duality in IPPs. The usefulness of the function h is clearly seen in the following IPP case: let us consider an IPP consisting of articles as sources and of authors of these articles (possibly multi-authored articles) as items: i.e. considering articles with 1,2,3,... authors. f(j) is then the number of articles with j=l,2,3,... authors and h(s) is then the number of authors with a fractional score s = - in one paper. The function h has already been used in this context to model fractional j frequency scores for authors - see the papers Egghe (1993b) and Egghe and Rao (2002a) but in these papers the function h was determined approximately (in the continuous setting in Egghe (1993b)) or exactly but in the discrete setting in Egghe and Rao (2002a), hence a clear relation with the dual informetric theory (Section II. 1) did not exist. We will present it here for the first time. We have the following theorem, generally valid in informetrics (even nonLotkaian).

Theorem II.3.1 (Egghe, unpublished): For every j e [l, pm ]:

'('I f(j) = - ^

or, what is the same: for every s £ fo(0),l|:

(11.77)


147

f (I| h(s) = ~ ^

(11.78)

Proof: We start by the formula (II.7):

°(i) = ^

)

ie[0,A]. Hence CT' = —

,/.\

N dr

dp,™

1

CT(l)=

T(T^)d7(T-r)d[

since we defined p ( A - i ) = p ( T - r ) iff A - i = V ( T - r ) , see (II.6). Since g(r) = p ( T - r ) (by (II.8)) we have

_,m U

=

1

dg(r)dU(i)

g2(r) dr

di

since r = U(i) because this is equivalent with A — i = V (T — r) by (II.3). But

dU(i)

,,

1

1

1

^ = 0 i )=—-i—-=— r = -rr U p(A-i) p(T-r) g(r) di hence

(11.79)

148


tf'(i) = - l ^

(H.80)

\G' = -£-, g' = - ^ .Hence, for every j e [ l , p m ] : 1, di dr)

i_

~~W) since j = g(r) (definition of g). Hence, by (11.80):

S{!)

=WWV)

since r = U(i) and s = o(i) (definition (11.76)). But, by (11.79)

'=°®=W)=T (11.82) and (11.76) in (11.81) yields

hence (11.77). From this (11.78) readily follows from (11.82).

(IL82)


149

Note that, up to a constant, formula (11.78) already appears in Egghe (1993b) but where the

f

A)

formula is obtained via approximations. The constant m, = — appears in Egghe (1993b) since there one works with fractions (hence normalized in [0,1]) while here we have functions on r £ [0,T] and on i £ [0, A] which incorporate this constant. In any case it is remarkable that the approximate argument in Egghe (1993b) produces an exact result.

In Lotkaian informetrics we have the following corollary to Theorem II.3.1, hereby completing the equivalencies proved in Theorems II.2.2.1, II.2.2.3, II.2.2.6 and II.2.2.7.

Corollary II.3.2 (Egghe, unpublished): The following assertions are equivalent:

(i)

je[l,p m ], C > 0 , a > 0 constants

(ii) h(s) = Cs"-3

(11.83)

s £ [CT (0), l], C > 0, a > 0 constants.

Proof: This follows readily from (11.77) and (11.78) but can also be proved directly from definition (11.76), using the function a and Theorems II.2.2.1, II.2.2.3, II.2.2.6 and II.2.2.7.

Note that the power-type law of f (Lotka) implies a power-type law of h, and vice-versa.

150


II.4 THE PLACE OF THE LAW OF ZIPF IN LOTKAIAN INFORMETRICS II.4.1 Definition and existence

In Section II.2 it is shown that, if a > 0 and o ^ l w e have equivalency of the law of Lotka

f(j) = ^

(H.84)

j € [l,p m ], and the law of Mandelbrot

8W

=07F7

(II 85)

-

r e [ 0 , T ] , with E = p m ,

F= ^ Cpm

(H.86)

P= ^ T oc-1

(H.87)

Definition II.4.1.1: The law of Mandelbrot for F = 1 is called the law of Zipf.

This conforms with the classical law of Zipf since, if F = 1 we have

.(0=^ r € [0,T], hence the classical function

aim


g(r') = ^

151

(H.89)

r1 > 1 (note that r1 € [l,T +1]).

Can F = 1 be realized? It is clear that this requires, by (11.86) that

a = l + Cpâ>l

(11.90)

for all a . This implies, by (11.87) that Zipf s law is limited to the case (5 > 0 (contrary to the (3 in Mandelbrot's law), which is also the case in the classical definition. Equation (11.90) is the necessary and sufficient condition to have Zipf s law and determines implicitely the place of Zipf s law in Lotkaian informetrics. Indeed, the above shows that a part of Lotkaian informetrics is Zipfian. Reversely, if we have Zipf s law, we have Mandelbrot's law with F = 1 and hence Lotka's law (11.84).

If we have Zipfs law, Lotkaian informetrics is automatically limited (by (11.90)) to c o l . But for every a > 1 we have a case where Zipfs law applies, namely if (11.90) is satisfied, which can always be achieved: take

i

Pm = - ^ r ct-1.

(H.91)

Note that (11.91) produces a pm > 1 (as required) as the following argument shows: By (11.55) we have

cpir But (11.34) implies

152


Hence (11.55) now becomes

F= — — a-1

(11.92) T

The condition F = 1 (for Zipf) implies

T=— a-1

1

(11.93)

Q

hence, since T > 0 we have that

> 1 and hence, by (11.91), pm > 1, since a > 1.

II.4.2 Functions that are equivalent with Zipf s law

From the general Theorems II.2.2.1 and II.2.2.3 we can deduce the functions that are equivalent with Zipf s law: as explained above, only the values a > 1 are possible.

Theorem II.4.2.1 (Egghe (1985,1989,1990a), Rousseau (1988,1990a)):

(i) f(j) = ^- (Lotka'slaw)

je[l,p m ],

P

=

f C]^

U-i.

Basic theory ofLotkaian informetrics 153 (ii) ( Z i p f S laW itSelf)

8 ( r ) = T^J

r€[O,T],E = p m , p = - i

(iii)

T

.

If a * 2:

"("-^-(f-^'f' i

r e [0,T], again with pm = - ^ j

.

If a = 2 :

G(r) = Cln(l + r)

(11.94)

re[O,T], the special form of Leimkuhler's law (b = 1) which can also be considered

as a

special

form

of Brookes'

G(r) = cln[d(l + r)] withd = l.

(iv)

If a * 2

i

i-\

{ 2-0. . 2 - a V - 2

1

f C I"1? i € 0, A , again with p = lot —lj

law (Brookes

(1973):

154

Power laws in the information production process: Lotkaian informetrics If a = 2: Bradford's law:

CT(i) = (lnK)K'

(11.95)

i € [0, A], K = e c > 1, being equivalent with the formula

U(i) = K ! - l

(11.96)

(graphical form of Bradford's law).

Proof: This follows readily from the results in Theorems II.2.2.1 and II.2.2.3 and the fact that the original (also called verbal) form of Bradford's law (the function a ) and the graphical form of Bradford's law (the function U) relate as

U(i) = / o i oo if and only if pm —> oo.

Basic theory of Lotkaian informetrics

155

Proof: Formula (11.34) says

a —1v

'

Hence, by (11.91) we have

T = P r i — i r =Pm"'-i

(n-97)

Pm

and, as always when Zipf applies: a > 1 (see (11.90)). Hence T —> oo iff p m —> oo.

This result is not true for general Lotkaian systems. Indeed, as expressed by equation (11.37), p m only depends on a and \x = — .So if we let T and A go to oo such that \i is constant, and if we keep a fixed, we have that pm is fixed. A concrete example is given in the first example of II.2.1.2.3: for n = 1.5 and a = 1.5 we calculated that pm =2.251. There T = 10,000 and A = 15,000, but any common multiple of these values (hence for A,T —> oo) yields the same pm < oo .

From the above (if Zipfs law applies) and Proposition II.2.1.1.1 it follows that T = oo implies pm = oo, hence a > 2 , if A < oo. Hence, by (11.87), (3 < 1. So, Proposition II.2.1.1.1 is only valid for Lotka's law and not for Zipfs law (if it were valid then T = oo implies P > 2, which is false, as expressed above).

The above result (T —> oo iff pm —> oo in case Zipfs law applies) will become crucial in Chapter IV where we can, because of this result, develop concentration theory for Lotka's law and Zipfs law at the same time.

2.

Note that it follows from (11.97), (11.88) and the fact that E = p m and (3 = a —1 (Theorem II.4.2.1) that

156


g(r) =

[

(11.98)

Vre[O,T].

3.

Because Zipf s law is a special case of Mandelbrot's law and because the latter one is completely equivalent with Lotka's law one is inclined to think that Zipf s law is not equivalent with Lotka's law. It is true that not all C and pm -values for Lotka's law imply Zipf s law (as is clear from the above) but, for the mentioned parameter values (i.e. (11.91)), we have equivalence of any Lotka power law with exponent a > 1, with any Zipf law with exponent P =

. a —1

4.

Note that, by (11.96), U(i) = K1 - 1 and not K ! , the latter one being the function in the "classical" graphical formulation of Bradford's law (in the group-dependent version see Wilkinson (1973), Narin and Moll (1977) or Vickery (1948)). It turns out that the pure exponential form of U can never be equivalent with the pure exponential form of G , contrary to the historical formulations - see also Egghe (1991).

Ill THREE-DIMENSIONAL LOTKAIAN INFORMETRICS

III.l THREE-DIMENSIONAL INFORMETRICS In Chapter I we introduced the informetrics theory by describing sources (the "producing" objects), items (the "produced" objects) and a matching function describing which source(s) produce(s) which items. In this connection we could talk of two-dimensional informetrics (sources/items or type/token) which is more than twice one-dimensional informetrics in which only data with respect to the sources, respectively items are studied (e.g. number of sources, number of items): in two-dimensional informetrics one also studies and quantifies which sources produce which items (the matching function) and yields, as a consequence, the informetric laws (functions).

Given an item set in two-dimensional informetrics we can determine different sets of sources that produce these items. Indeed e.g. for articles as items one can consider journals (in which these articles appeared) as sources but one could consider the authors of these articles as sources as well. Note that, in this chapter, we do not address the problem of multiple authorship: we will deal with this (intricate) problem in Chapter VI; in the present chapter we deal with different source sets such as journals and authors producing articles. Reversely (or in a dual way) we can consider a source set and determine two item sets of produced objects by these sources. An example is given by articles giving references but also receiving citations. Note that in the first example an article was an item while in the second example an article is a source. This leads us automatically to a third case: the case of two source sets and two item sets but where one source set is equal to one item set. Example: the source set of journals produces the item set of articles in these journals. This item set is then considered as our second source set producing a second item set, e.g. of references in these articles. Such

158


systems are the object of study in three-dimensional informetrics. Again also here, threedimensional informetrics is more (is a "higher" discipline) than three times a one-dimensional informetrics theory or than two times a two-dimensional informetrics: the different matching functions are not independent from each other and, hence, their relations must also be studied. From the above introduction it is clear that we can determine three types of three-dimensional informetrics. Let us now describe them in more detail (cf. also Egghe (1989, 1990a) and Egghe and Rousseau (1990a)).

III. 1.1 The case of two source sets and one item set

This case can be symbolized by Si, S2 the two source sets, I = the item set and depicted as in Fig.III.l

Fig. III. 1

Three-dimensional informetrics: two source sets and one item set (—> means: produces).

The most classical example is the one given above: Si = {authors}, S2 = {journals}, I = {articles}, where we consider the authors and journals as producers of articles. Other example: I = {articles} produced by Si = {authors} but also by S2 = {research institutes}, i.e. the research institutes where these authors are employed. Of course, this example could also be considered in the "linear" way (see Subsection III. 1.3): institutes have authors and authors have articles. Of the same (double) nature are the examples of Lafouge (1995, 1998): volumes (of journals) contain articles and these articles are used (e.g. number of (inter)library requests). Although this example could also be classified in the coming Subsection III. 1.3, Lafouge also considers the "triangle" form (as in Fig.III.l) with two source-sets and one itemset. In our notation, his triangle looks like in Fig.III.2 (g,, g2 are the two rank-frequency

Three-dimensional Lotkaian informetrics

159

functions). Here h is the mapping of which article (in Si) belongs to which source (in S2) and hence the link between the two IPPs is made via the symbolic formula

g2(r2) = X-, (r2) g.( r .) dr ,

Fig. III.2

On-i)

Linking of two IPPs in the case of Fig.III.l (p r a l and p m 2 denote the maximal item densities p m in a source in the two IPPs - see Subsection II. 1.1).

(Lafouge uses a discrete setting and hence a 52 instead of an / ) . In this sense, use of the volumes is determined by a cumulation of the use of the articles that belong to these volumes. Lafouge then employs this model to determine size-frequency distributions for volume-use based on (non-Lotkaian - e.g. geometric) size-frequency distributions for article-use.

By duality, i.e. replacing items by sources and sources by items (i.e. replacing "produces" by "is produced by") we have that the above case is essentially the same as the next one on which more examples are known.

III.1.2 The case of one source set and two item sets

This case can be symbolized by S = the source set, Ii, h the two items sets and depicted as in Fig.III.3.

160


Fig. III.3

Three-dimensional informetrics: one source set and two items sets.

Examples: Papers have references and receive citations, so papers produce references and produce citations; authors produce articles and produce article pages. Journals publish articles but also contain authors. So authors do not always have to be considered as sources (cf. the Subsection 1.4.4 devoted to authors as items of articles, i.e. where one considers the number of authors per paper). In this connection we also have the example: articles producing authors but also citations. From Coleman (1992) we obtain the example: labs are sources of scientists and of articles. In Yitzhaki (1995) one has the example of articles as sources for pages (or another measure of the length of an article such as words or even characters) and of references hereby also studying the relation between the length of an article and the length of its reference list. Kyvik (1990) studies authors as sources of articles and of ages. A bit strange is the following example that is implicite in Qin (1995) but stated explicitely in Egghe and Rousseau (1996c): authors as sources of publications and of co-authors (the latter variable for Lotka's function has also been studied in Gupta and Karisiddippa (1999)).

Although we have many examples, not many results are known in this case. We refer the reader to Egghe and Rousseau (1996c) for a first attempt to derive three-dimensional properties in this case. The results, however, are not far reaching, leaving open the general problem to describe the situation depicted in Fig.III.3.

Here we may give one indication on how one may proceed. Let g[ respectively g2 be the rank-frequency functions of the two IPPs (S,Ii) and (SJ2) respectively. As we indicated already, we must find the "added value" of three-dimensional informetrics above the description of these two-dimensional IPPs. It is clear from Fig.III.4 that the function


k= g2V

161

(ni.2)

shows the link between these two IPPs, hereby indicating the link between the items in Ii and the items in I2, being produced by the same source (that is why the rank-frequency functions g, and g2 are used and not the respective size-frequency functions, as was done in Egghe and Rousseau (1996c)).

Fig. III.4

Linking of two IPPs in the case of Fig.III.3 (p m l and p m 2 denote the maximal item densities pm in a source in the two IPPs).

Supposing power laws for g, and g2 (i.e. Zipf s law), it is obvious from (III.2) that also the function k is a power law and this might be regarded as a first (elementary) result on this type of three-dimensional informetrics.

III.1.3 The third case: linear three-dimensional informetrics

This case is, by far, the most important one and refers to the case of two source sets Si and S2 and two item sets Ii and I2 but where the second source set S2 equals the first item set l\. We can depict this situation as in Fig.III.5.

s,

Fig. III.5

>

i,-s2

>

i2

The case of linear three-dimensional informetrics: the item set of the first IPP is the source set of the second IPP.

162


This means that the items in the first IPP become sources (producing items) in the second IPP. Fig.III.5 makes it clear why this theory is also called linear three-dimensional informetrics.

Examples: Researchers (sources in Si) produce articles (items in Ii) which become sources (in S2= Ii) of citations (or references) (items in I2). Of the same type is the "chain": journals —> articles —» citations. In Rousseau (1990b) one studies the linear three-dimensional problem: journals —> articles —> software programs (discussed in these articles). Linear three-dimensional informetrics problems were also (implicitely) mentioned in Rousseau (1992a) and Fox (1983). Rousseau (1992a) gives the example of a public library where one has categories of music, each of them having CDs available, being items of these categories but also sources of their borrowings. Of course many more examples can be given: the example in Subsection III. 1.2 of Coleman (1992) can be rephrased: labs (or countries) are sources of scientists (items) which become sources of articles. Also: authors produce papers and these papers have pages.

Unique for this case is (as is clear from Fig. III.5) that a "new" IPP is constructed, namely the IPP with source set Si and item set I2. A natural question arises: describe the rank- and sizefrequency functions of the "composed" IPP (S1J2) from the two IPPs (Si,Ii), (S2J2) (with I] = S2 of course). This will be done in the next Subsection III. 1.3.1 under the name "Positive Reinforcement" indicating the reinforcement of the production of the sources in Si via the composition with the IPP (Ii = 82,12). This problem will be completely solved for the (only important) case that the source-rankings r2 on S2 in the second IPP (S2J2) are the same as the rankings in S2 = Ii, induced by the source-rankings n in the first IPP (more explanation in Subsection III.1.3.1).

A bit more veiled but at least as important is another study that is possible in the case of Fig. III.5: Type/Token-Taken informetrics. In this theory (to be developed in Subsection III. 1.3.2) the informetrics of the use of items is described (instead of their existence). More details will be given in the sequel with many practical examples. Type/Token-Taken informetrics boils down to the study of the second IPP (S2J2) but where (as in Fig. III.5) S2 = Ii (and the influence of the IPP (S1J1) is incorporated) and where we keep the same item set: I2 = Ii (compare with positive reinforcement where the source set (Si) is kept fixed but where I2 is the item set). In this connection, we can consider Type/Token-Taken informetrics as the dual


163

of the positive reinforcement theory - see the next two subsections for more details).

III.1.3.1 Positive reinforcement

Positive reinforcement of an IPP is a feature that can be defined independently from threedimensional informetrics as the next definition shows.

Definition III.1.3.1.1: Let S = [0,T], I = [0, A] be an IPP with rank-frequency function g. Let I* = [0, A*]. We say that the IPP (S,I*) is a positive reinforcement of the IPP (S,I) if its rank-frequency function g* is given by g'=9°g,

(HI.3)

where (p is a strictly increasing function such that cp(x)>x for all x. By definition of g (II.8), the same relation is true if g is replaced by p (since T and r are the same in both IPPs).

The interpretation of the above definition is clear: the positively reinforced IPP (S,I ) has the same sources but with an increased productivity, given by (III.3).

We have the following basic result on linear three-dimensional informetrics, showing that the composed IPP (S1J2) (as described above) is a positive reinforcement of the IPP (Si,Ii), at least under the following (completely natural) restriction, which is the only important case of linear three-dimensional informetrics.

Restriction HI.1.3.1.2: We will restrict ourelves to the case that the source-rankings r2 in the second IPP are the same as the rankings in Ii, induced by the source-rankings r, in the first IPP. We will express this restriction in a mathematically exact way but first we give an intuitive interpretation: items in Ii are ranked according to the ranking we have in Si (as always, sources are ranked in decreasing order of production). Considering Ii = S2 as sources in the second IPP, we require that their productivity in the second IPP is such that the source-ranking r2 in this second IPP is the same as the one we have already in Ii. An exact formulation on this restriction is

164


r 2 = / o r ' g , ( r ) dr

(III.4)

Note that, by (II.9) and (11.18), r2 is nothing else than G(r,) = IT 1 (r,) in the IPP (Si, Ii). Note also that (using (II.3)) r2 G It = S 2 , as it should.

Theorem IH.1.3.1.3 (Egghe (2004e)): Let the first IPP have source-set S, = [0,T,], item-set I, = [0,A,], size-frequency function fi and

rank-frequency

function

g,

and

let

the

second

IPP

have

S2 =[0,T 2 ] = I, =[0,A[], item-set I 2 =[0,A 2 ], size-frequency function

f2

source-set and rank-

frequency function g 2 . Then the composed IPP (under restriction III. 1.3.1.2) has source-set S = S,, item-set I = I2 and rank-frequency function

g = j , we hence have cp(j,) > j , for all j , . That (p strictly increases follows easily from (III.5):

J

3 .

Hence, if a < 3 and since A = /

jf (j)dj < 00 (this boils down to a > 2 - see Proposition

178


11.2.1.1.1) we have, by (III. 15), that n* = oo. If a > 3 we have by (III. 15) and (111.29)

Using Proposition II.2.1.1.1 (existence theorem in case pm = oo) and formulae (11.28), (11.29) we can conclude:

T u. =

1 =

2T-A

2-n

and T and A are finite since a > 3 > 2.

Note IH.2.2.2: Note that (111.28) implies \i > \i, a fact that has been generally proved in Theorem III. 1.3.2.1. Note also that the relation (111.28) is limited to \i € ]l,2[, since A > T (hence n > 1) and since a > 3 implies, by (11.30), that \i < 2 . We hence have the functional relationship of p.* in function of \i as depicted in Fig. III.7. Note that the graph is tangent to the straight line y = x which visualizes the fact that \x* > \i. for \i > 1.

Fig. III.7

The function \i (JJ.) .

Three-dimensional Lotkaian informetrics 179

Now we will treat the general case. Here no direct formula between \x and ji* exists but we are able to prove a formula for n* in function of a parameter x which is the solution of an equation (in which (x appears) which is also given here and solved numerically. So the relation |x* (|i) can always be calculated.

We suppose (111.27) to be valid for j e [ l , p m ] ; see also Subsection II.2.1.2 for its existence. From this theory it followed that: from (11.20), (11.21), (11.34) and (11.35): if a > 1

T = J| P "'f( J )dj = T ^ - ( p ^ - l )

(111.30)

A=j;p"jf(j)dj=^-(Pr-i)

(HI.31)

and if a ^ 2

If a = 2 we have (see (11.36))

A = J ^ j f ( j ) d j = Clnp m

(111.32)

From this it followed that (cf. (11.37)) if a ^ 2

^ = A = iz£L i £ z l T

2 - a pL--l

and if a = 2 (cf. (11.38)):

Pm

(in.33)

180


To calculate \x* we can argue: if a =*= 2, a * 3, then

J>U)dj =^(pL-a-l)

(IH.35)

jj P °j 2 f(j)dj = C ( P m - l )

(111.36)

j] P "j 2 f(j)dj = Clnp m

(111.37)

If a = 2 then

If a = 3 then

as is readily seen. Hence from (III. 15) it now follows that, if a =e 2, a ^ 3, using (111.31) and (111.35)

»

=

^

^

(in.38)

If a = 2 we use (111.32) and (111.36) to obtain

[i = ^ ^ -

(111.39)

If a = 3 we use (III.31) and (111.37) to obtain

u* = - ^ i j t_ Pm

(111.40)

Since neither (III.33) nor (111.34) can be solved exactly for pm and since p m appears in the formulae for ja* it is clear that a direct relation between n and \i is not derivable. But we


181

can solve (111.33) and (III.34) numerically which will yield indirect but concrete relations between \x and \x*. This will be done at the end of this subsection but first we prove an "exact" result.

Theorem III.2.2.3 (Egghe (2003)): For each a fixed we have that \x* is an increasing function of \i.

Proof: Denoting x = p m > 1 in (111.33) and (111.38) we have the following equations:

=

2â 3-a

xâ-l x2-a-l

(a^2,a^3) V > ;

'

(111.42) v )

Let 1 < a < 2

Then u, increasing implies — increasing since < 0. Hence, by (111.41) (x a - 2 a —2 x2~a-l x'-a-l

decreases. By Lemma III.2.2.4 (below) (since 1 — a and 2 — a have opposite sign) we have that x increases. Again by the same Lemma we have that

x3-a-l x2-a-l

increases (since 3 — a and 2 — a have the same sign), hence by (III.42), \x" increases.

182


2

x3-a-l

2_a \=>n x —1

. „

/.

a>3

Now

„

x2"a-l

1 a-1 .

A

x

x3"a-l

„

, ,

a=2

From (111.34) and (111.39) we have

V = ^T 1-x

(HI.43)

H*=î lnx

(111.44)

- t - = J^—

= >0

since In x < x — 1 always (since ex > x + 1 , for all x > 1). Also


(lnx) 2

dx

since Inx > 1

183

. This follows from the fact that the function 9(x) = Inx -\ x

1 is minimal x

in x = 1 and 9 (l) = 0. Hence \i /"=> \x /*.

a =3

From (111.33) and (111.40) we have H= 2 ^ -

(111.45)

x+1

V=-^r1-x

(HI.46)

Obviously n strictly increases in x and by the above (j,* strictly increases in x. So

n/^nVLemma HI.2.2.4: The function

f(x) = ^ f ^

(111.47)

increases if a and a + 1 have the same sign and decreases if a and a + 1 have opposite sign.

Proof:

, , x a -'(x a + 1 -ax-x + a) f x = ^ z >[ }

(* a "l) 2

184


So f'(x) has the same sign as

cp(x) = xa+1 - a x —x + a.

Now

cp'(x) = (a + l ) x ' - ( a + l)

which is zero in x = 1. Furthermore

cp"(x) = a(a + l ) x a - \

so

cp"(l) = a(a + l ) .

So if a (a +1) < 0 then cp has a maximum in 1 and cp (l) = 0. So
0 then ep has a minimum in 1. So cp(x)>0 for all x and hence f increases.

As will become clear from the sequel (Table III. 1) we conjecture the following, but we are not able to find a proof for it.

Conjecture IH.2.2.5: For each fixed |a we have that \i is an increasing function of a .

Finally, we will calculate the TTT average |j* in function of the TT average \x. The calculation of (i* in function of \i and a requires the solution (for x) of the equation (cf. (111.41)), if a ^ 2


(n a - 2

(a

185

a-2

andof(cf. (111.43)), if a = 2

lnx + ^ - p , = O. x

(111.49)

Once x is found we have then jx* using the formulae: if a ^ 2, a =* 3 (cf. (III.42))

'=i_a

3-a

x

J_

x 2 -"-l

if a = 2 (cf. (111.44))

H*=4-^

(HI.51)

lnx and if a = 3 (cf. (111.46))

V T ^ 1-x

(111.52)

In Egghe (2003a) we solved (111.48), (111.49) for x using the MATHCAD 4.0 software package. Note, however, that a and [i cannot be taken fully independent of each other: there is a lot of freedom but we have the restriction (11.32) of Theorem II.2.1.2.1, if a > 2 :

H < ^

(01.53)

a-2 In the next table (of \x' in function of ]x and a ) we hence have the following restrictions

186


a = 2.5=>jinn< 1.6667

Table III. 1 Values of \x* in function of n and a . Reprinted from Egghe (2003), Table 1, p. 608. Copyright John Wiley & Sons Limited. Reproduced with permission.

JJ*

a = 1.5

a =2

a = 2.5

a =3

a = 3.5

H = 1.2

i- 2148

1.2151

1.2157

1.2164

1.2178

H = 1.5 ^=2

i- 5838 2.3317

1.5984

1.6177

1.6479

1.7010

2.4612

2.7320

-

-

3.2492

3.7279

5.8541

-

-

5.5996

-

-

8.3955

-

-

^ = 2.5 H= 3

^ = 3.5

4.3333 5.5894

Another way to produce (n,H*) -relations, that is simpler than the above method, is by inputting given values of x = pm > 1 and a > 1 into (111.48) (or rather (III.33)) or (111.49) (or (111.34)) for ]x and (111.50) or (III.51) or (III.52) for u*, so that no numerical solution as above is necessary. In this way we obtain Table III.2.

Table III.2 Values of n and |x* in function of x = pm and a .

M

a = 1.5

a =-2

a = 2.5

=3

a = 3.5

x=1.5

(1.225, 1.242) (1.216, 1.233) (1.208,1.225) (1.200, 1.216) (1.192, 1.208)

x=2

(1.414, 1.471) (1.386, 1.443) (1.359, 1.414) (1.333, 1.386) (1.309, 1.359)

x=3

(1.732, 1.911) (1.648, 1.820) (1.570,1.732) (1.500, 1.648) (1.438, 1.449)

x=4 x=5

(2,2.333)

(1.848, 2.164)

(1.714,2)

(1.600, 1.848) (1.505, 1.714)

(2.236, 2.745) (2.012,2.485) (1.821,2.236) (1.667, 2.012) (1.545, 1.821)

IV LOTKAIAN CONCENTRATION THEORY

IV. 1 INTRODUCTION Concentration theory studies the degree of inequality in a set of positive numbers. It is not surprising that the historic roots of concentration theory lie in econometrics where one (early in the twentieth century) felt the need to express degrees of income inequality in a social group, e.g. a country. Hereby one expresses the "gap" between richness and poverty. One of the first papers on this topic is Gini (1909) on who's measure we will report later.

The reader of this book will now easily understand that concentration theory takes an important role in informetrics as well. Indeed, as is clear from Chapter I, mformetrics deals with the inequality in IPPs, i.e. in production of the sources or, otherwise stated, the inequality between the number of items per source. As we have seen, Lotkaian informetrics expresses a large difference between these production numbers. Just to give the most obvious example: if we have Lotka's law (with exponent a = 2, just to fix the ideas): f(n) = — then n f (2) = ——, f (3) = ——, fw(4) = —— and so on, where fv(n) denotes the number of sources w V; 9 16 ' 4 with n items. It is clear that, expressed per production class n, there is a large difference between the number f (n) of sources in these classes. Zipf s law is also a power law, hence it also expresses a large difference but now between the numbers g(r), r = 1,2,3,..., where g(r) denotes the number of items in the source on rank r (where sources are ranked in decreasing order of their productivity). It is clear that all examples of sources and items, given in Chapter I, can be the subject of a concentration study. The skewness of these examples was apparent and hence one should be able to measure it.

188


Generalizing the above examples, we can say that we have a decreasing sequence of positive numbers x,,x 2 ,...,x N , N e N, and we want to describe the degree of inequality between these numbers, otherwise stated, the degree of concentration: a high concentration will be where one has a few very large numbers x,,x 2 ,... and many small numbers ...,x N _,, x N . It is clear that this must be formalized. We will use techniques developed in econometrics but we will also report on the "own" developments that have been executed in informetrics itself. Under the "own" developments we can count the so-called 80/20-rule and the law of Price. The main part of this chapter, however, will be the study of the Lorenz curve which was developed in econometrics around 1905 (cf. the historic reference Lorenz (1905)).

Let us briefly (and intuitively) describe these concepts here, before studying them more rigorously in the further sections. The simplest technique is the 80/20-rule which states that only 20% of the most productive sources produce 80% of all items. Of course, this is just a simplification of reality: it is the task for informetricians, in each case, to determine the real share of the items in the most productive sources: 20% of the most productive sources might produce 65% of all items but this could as well be 83.7%! Also, we do not have to consider 20% of the most productive sources: any percentage can be considered. So, generalizing, we can formulate the problem: for any x € ]0,l[ determine 0 e ]0,l[ such that 100x% of the most productive sources produce 1000% of all items. We can even ask to determine 6 as a function of x. This "generalized 80/20-rule" could be called the determination of "normalized" percentiles since both x and 0 belong to the interval [0,l] while in the calculation of percentiles, one of these numbers is replaced by actual quantities (of items or sources). Since both x and 0 denote fractions this technique is (sometimes) called an arithmetic way of calculating concentration (see Egghe and Rousseau (1990a)).

In this sense we can call the law of Price a geometric way of calculating concentration. The historic formulation (see De Solla Price (1971, 1976) and implicite in De Solla Price (1963)) i

states that, if there are T sources, the vT = T ^ most productive sources produce 50% (i.e. a fraction —) of all items. For evident reason, this principle is also called Price's square root law. It is clear how to extend this principle: let 9 6 ]0,l[, then the Te most productive sources produce a fraction 9 of all sources. This is called Price's law of concentration and we will

Lotkaian concentration theory

189

investigate in what cases in informetrics this is true. Also this principle could be generalized stating that for 8 E ]0,l[ the top TE sources produce a fraction 9 of all the items and we can ask for a relation between s and 0.

Both general formulations of the 80/20-rule (in terms of x and 0) and of the law of Price (in terms of e and 0) involve two numbers. We could wonder if we can construct a function F such that, for any decreasing vector X = (x,,x 2 ,...,x N ), with positive coordinates, the value F(x) = F(x,,...,x N ) is a good measure of the concentration in X. It is clear that an "absolute" good value for F(X) does not exist but we can determine requirements for the value of F(X) in comparison with values F(X') for other vextors X' as above, i.e. to give relative valuejudgements. Let us formulate some "natural" requirements.

(i)

F(X) should be maximal for the most concentrated situation, namely for a vector X of the type X = (x,0,.. .,0) where x > 0.

(ii)

F(X) should be minimal for the least concentrated situation, namely for a vector X of the type X = (x,x,.. .,x) where x > 0.

In terms of wealth or poverty, (i) states that X = (x,0,...,0) must have the highest concentration value (given F), since one source (e.g. person) has everything and the other sources have nothing. Condition (ii) states that if everybody has the same amount (e.g. money), the concentration value should be minimal (and preferably zero).

(iii)

F(X) should be equal to F(cX) where, for X = (x,,...,x N ), the vector cX is defined as (cx,,...,cx N ), forallc>0.

Condition (iii) is called the scale-invariant property and is requested since describing the concentration of income (i.e. describing wealth and poverty) should be independent on the used currency (€, $, Yen,...) which all are interrelated via a scale factor. The next property is also very important:

190


(iv)

F(X') > F(X) if X' is constructed from X by "taking (in X) an amount away from a poor person and giving it to a rich person". In other words, we require F(X') > F(X)incase X = (x,,...,x N ) and X1 = (x,,...,x i +h,..., X j -h,...,x N )

(IV.l)

if 0 < h < x r Condition (iv) is called the transfer principle and was introduced in 1920 by Dalton - see Dalton (1920). It is clear that the transfer principle is a very natural requirement and in Egghe and Rousseau (1991) we showed that it comprises properties such as

(v)

"If the richest source gets richer, inequality must rise": for all X = (x,,...,x N ) (as always, ordered decreasingly) and h > 0 we have that, if X' = (xj +h,x 2 ,...,x N ), then f(X') > f(X). The same can be said if the "poorest source gets poorer",

(vi)

The principle of nominal increase: if, given a vector X = (x,,...,x N ), the production of each source is increased with the same amount h > 0, inequality is smaller: for X' = (x, +h,...,x N + h ) we have f(X!) < f(X).

We have, deliberately, used the econometric terminology, just to illustrate these principles but it is clear that these principles are universally required in any application of concentration theory.

A genius invention of Lorenz, however, requires the construction of a Lorenz curve (denoted L(X)) of a given decreasing vector X and these Lorenz curves have the property that any measure F satisfying L(X) < L(X')=>F(X) < F(X') automatically satisfies all the above principles. In words: any function F on vectors X = (x,,...,x N ), which agrees with the Lorenz order, is satisfying the above requirements for concentration measures and hence can be called a "good" concentration measure. The Lorenz curves have, in addition, the property that any generalized 80/20-rule can be seen on these curves, hence they comprise this aspect of concentration theory as well.


191

Therefore, in the next section we will develop Lorenz concentration theory for vectors X = (x,,...,x N ) as above. Then we will check for the existence of such good concentration measures and we will describe their properties. We are then at the challenge of developing Lotkaian concentration theory, i.e. to calculate Lorenz curves and good concentration measures if the vector X is of power type, i.e. X = (f(l),f(2),...,f(n mas ))

(IV.2)

where f is Lotka's size-frequency function (nmax is the maximal production of a source) or if

X = (g(l),g(2),...,g(T))

(IV.3)

where g is Zipf s rank-frequency function (T is the total number of sources). We will see that exact calculations of these discrete versions of the laws of Lotka and Zipf are, also in this application, not possible. As explained in Chapter I, also here we will encounter problems of evaluating discrete sums, which is not possible. The previous three chapters fully showed the power of the use of the continuous versions of the laws of Lotka and Zipf, i.e. the functions (11.27) and (11.89). But we face here the need of extending Lorenz concentration theory from discrete vectors X = (x,,...,x N ) to continuous functions h on an interval [l,x m ]: for h = f (Lotka's function) we have xm = p m and for h = g (Zipf s function) we have xm = T + 1 (see Chapter II). Even for a general (non-Zipfian) rank-frequency function g, defined on [0,T] (see II.8) we can apply such a theory on the interval [l,T + l] by replacing the re[0,T] by 1 + r e [l,T +1] as we did for Zipf s law. If we can extend the Lorenz concentration theory to functions h : [l,xm] —> M+ , we can apply it to the functions f of Lotka and g of Zipf in an identical way since both f and g are functions like h. This will be done from Section IV.3 on: based on our insights in the discrete case we will define the Lorenz curve L(h) of a general function h on an interval [l,xm] and determine its special form if h is a power law. Our approach will be simpler than (but equivalent with) earlier approaches e.g. of Atkinson (1970) and Gastwirth (1971, 1972). We will determine three important good concentration measures based on L(h), namely the Gini index, the coefficient of variation and Theil's measure and calculate their values for power laws h. These results can then be applied to the power laws of

192


Lotka (f) and of Zipf (g) and we will determine the crucial role of the exponents a and P in this matter ( a in (11.27) and P in (11.89)). We will be able to present concrete formulae for the mentioned concentration measures in function of a and P, hereby proving concentration properties in function of a and p . This is the real heart of Lotkaian concentration theory. From this Lorenz concentration theory we will also determine the generalized 80/20-rule in Lotkaian informetrics and we will also show that Price's law of concentration is (exactly) valid if we have Zipf s law.

We close this introduction with an important remark: although, once Lorenz concentration theory is developed for general power functions h, we can apply it to f (Lotka) and g (Zipf) in a mathematically identic way, the interpretation of concentration theory on f is completely different from the one on g. The latter theory is the most important one since it describes the inequality among the sources (expressed by their ranks r) with respect to their item production. It is this application that is always studied, also in econometrics. But, as said, in a mathematical equivalent way, we can study concentration aspects of the function f, hereby calculating the inequality between the different source productivity levels j (as in (11.27)). This is a functionality that - as far as we are aware of- only occurs in informetrics: Zipf s law also occurs in linguistics, econometrics, physics,... but Lotka's law is a regularity that is only studied in our field. Because of the fact that it comprises Zipf s law and even Mandelbrot's law (Chapter II) and the fact that Lotka's law is a simple power function, we think it plays a central role in informetrics (hence the writing of this book!) and hence we think it is worthwhile to study also concentration properties of Lotka's function, in addition to the study of the concentration properties of Zipf s function.

IV.2 DISCRETE CONCENTRATION THEORY In order to describe the concentration (inequality) of a vector X = (x,,x 2 ,...,x N ) we have to introduce the Lorenz curve as a universal tool in the construction of a wide range of concentration measures. We will suppose that all xt > 0 (although extension to the case that some X; are negative is possible - we will not go into this since we do not need it in this book) and that X is decreasing (this can always be achieved: if X is not decreasing we can reorder it decreasingly).


193

The Lorenz curve of X denoted by L(X), is constructed by linearly connecting the points (0,0) and

^,£ a j

(IV.4)

i = 1,...,N, where

aj=-j£-

(IV.5)

Note that the last point (for i = N) is (1,1). Since X decreases we have that L(X) is a concave polygonal curve that increases from (0,0) to (1,1). Its form is depicted in Fig.IV.l.

Fig. IV. 1

General form of a discrete Lorenz curve.

The power of this genius construction lies in the fact it describes all aspects of concentration in one graph: the higher L(X), the more concentrated X is. Let us illustrate this. The vector of "no" concentration is X = (x,x,...,x) with x > 0 and L(X) is the straight line connecting (0,0) with (1,1), the lowest possible curve (since a Lorenz curve is concave). The vector of

194


"highest" concentration is x = (x,0,...,0) with x > 0 and L(X) is the curve connecting (0,0) with —,1 and —,1 with (1,1), the highest possible curve. It is easy to see that if X' is

IN

J

INJ

constructed from X via an elementary transfer as in (IV. 1), we have that L(X') > L(X) (here > means > and not equal as curves). A theorem of Muirhead (see also Egghe and Rousseau (1991)) says that, conversely, L(X') > L(X) implies that X' can be derived from X via a finite number of elementary transfers (as in (IV.l)). Further, we trivially see that cX = (cx,,...,cxN) (c > 0) has the same Lorenz curve as X. So, the Lorenz curve is the right tool to describe concentration. It is now clear that any function C that respects the Lorenz order, i.e. which satisfies, for all vectors X, X'

L(X)( a i )(a;) i=1

i=l

(IV.8)


195

for all convex continuous functions (p. Here X = (x,,...,x N ), X ' = (xj,...,x N ) and (a,,...,a N ) and (aj,...,a N ) are defined as in (IV.3). From this result, applied to cp(x) = x 2 we have that

V 2 (X) = N ^ a f - l

(IV.9)

is a good concentration measure. Note that (IV.9) is equivalent with (by (IV.5)):

V2(X) = N £

-^-T-1 j=l

V

' u2 N t f '

K

V 2 (X) = - ^

(IV. 10)

being the quotient of the variance and the square of the average of X. For this reason one calls V = V(X) the variation coefficient and, because of the above, it is a good concentration measure. If we take cp(x) = xlnx we find Theil's measure (Theil (1967))

N

Th(X) = lnN + ^ a i l n a i

(IV. 11)

i-i

For other good concentration measures we refer the reader to Egghe and Rousseau (1990a, 1990c, 1991).

So far for general discrete Lorenz concentration theory. Since we want to have a Lotkaian concentration theory, we are at a point to interpret L(X), G(X), V2(X) and Th(X) for X as in (IV.2) or (IV.3). This is not easy for two reasons. First of all the linear parts in L(X) destroy the power law and make calculations on L(X) (e.g. for G(X)) difficult. Secondly, as already remarked in Chapter I, evaluating discrete sums as in (IV.9) and (IV. 11), ending up with

196


analytic formulae, is not possible. Using continuous functions is a solution to both problems: if we can define a Lorenz curve for a continuous function, we may fully work with this function and we do not have to introduce linear parts and also, finite sums are replaced by integrals which can be evaluated. This will be worked out from the next section on.

In Egghe (1987b) we made an attempt to calculate G (or rather Pratt's measure

N

G

N —1 which is close to G for large N and which will not have a purpose in our continuous theory to follow) for some informetric functions including the ones of Zipf and Lotka. The results are, necessarily, approximative. Since we will obtain exact results in the continuous case, we will not go into these results here.

IV.3 CONTINUOUS CONCENTRATION THEORY IV.3.1 General theory

Let h:[l,xm]—->R+ denote any positive decreasing continuous function on the interval [l,x m ], where xm > 1. Based on the discrete case we define the Lorenz curve of h, denoted L(h) as given by the set of points, for x e [l,xm]

x _!

JLj

X

In other words, putting y = x

fh(x')dx'

r,J'

(IV.12)

n,-! J ^ h ^ d X 1

e [0,1] (hence x = y(xm - 1) + 1), the Lorenz curve L(h) of m ~1

h is the function

r(x-1)+1h(X')dx' L(h)(y)=JY. J h(x')dx'

(IV.13)


197

This approach on defining the Lorenz curve is direct (on h) and similar to the definition of L(X) for discrete vectors X. It is different (in its formulation) than the definitions in Atkinson (1970) and Gastwirth (1971, 1972) in econometrics but all definitions are mathematically the same. In addition, our approach allows for finite xm , which is not the case in the other approaches.

It follows that

L h

( )'(y)=

r

*mr!

I

h(y(x m -i)+i)

(iv.i4)

hfx'jdx'

and that L(h)"(y)= j X m " l ) 2 h'(y(x m -l) + l) f h(x')dx'

(IV.15)

J1

hence L(h) is a concavely increasing function from (0,0) to (1,1) (since h > 0, h' < 0). Its general form is depicted in Fig.IV.2. Since L(h)' is continuous we have that L(h) is a C1 (i.e. a smooth) function (see e.g. Apostol (1957) or Protter and Money (1977)).

Fig. IV.2

General form of a continuous Lorenz curve.

198


As in the discrete case such a Lorenz curve can be used to measure the concentration in the (continuous) set of values h(x) for x e [ l , x m ] . Based on the discrete results, we can also define C to be a good measure of (continuous) concentration if we have the following implication for all functions h and h' as above:

L(h) oo is

lim G(h) = —

1 = —5L-

(IV.30)

but here 7 is restricted to 7 < 1. This is in accordance with the calculation of G(h) using (IV.26) which is also restricted to 7 < 1 and which also yields (IV.30), as can readily be checked.

So, for Lotka's function f we have, if pm < 00 :

G(f) =

' (p a -i)(pL--i)[^ p ^- 1 )- ( p -- 1 ) ]- 1

(IV31)

and for Zipf s function g we have (if T < 00 )

G(g) in function of a yields (using (IV.27))

G

(g) = -7

2

-l^—^ 7 ^ f ( T + 1 ) ^ - l "T "I

(IV.33)

T(T + l p - l H 2 a - 3 ^

For the limiting values we have for g, based on (IV.30):

limG(g) = —2— TÔC

w

(IV.34)

2-p

= —-— 2a-3

(IV.35) V

;


207

((3 2 ) . Result (IV.35) was already proved in Burrell (1992b), where only this limiting case (and a > 2) is considered. These results are in agreement with Corollary IV.3.2.1.5, as they should be. We leave it to the reader to calculate G(h) for y = 1, using formula (IV.25).

We refer to Egghe (1987b) for some discrete calculations of Pratt's measure (Pratt (1977)) which is essentially the same as the Gini index (see Carpenter (1979)).

IV.3.2.2.2 Calculation of the variation coefficient We present two methods: one based on (IV. 18) for general h and one based on the formula V = —, which only yields V for Zipf s function g.

1. First method By (IV. 18)

V2(h) = / o 1 [L(h)'f(y)dy-l,

where L(h) is given by (IV.24) (y * 1) and (IV.25) (y = 1). If y * 1, we have

L(h)'(y) = , I T ( 1 1 7 ) | X " " , \ ) ,V K Y -i)(y(x m -i)+i)

(IV36)

If y = 1, we have

L(h) (y)

' >o(;k'- 1 ) + 1 )

Let now y *= 1, y *= - . Then, by (IV.36)

( h )= f

2dy-i

^^(Hi'k-r-i).,

}

(x--l) 2 (l-2Y)

If y = —, then we have V2(h)=^m~^lnX--l

(IV.39)

4(V^-l) If y = 1, then (IV.37) implies

(x - i f 2

Vm

/ 1 (IV.40) (inx ) x So we have for Lotka's function f, if p m < oo (only the case a v= — ,a ^ 1) is given) V (h)=

(l-ay(p.y-l) (p^-l)(l-2a)

and for Zipf s function g, if T < oo and (3 ^ — ,p * 1:

(i-p)2T((T+iy-2f>-i) 2

V (g)= l j

;

lV

^ '—I ((T+ l ) - - l ) 2 ( l - 2 P )

which takes the following form in a > 1, a ^ 2, a ^ 3

(IV.42)


(a-2f

209

T T 1

[( + F - l ]

V 2 V( g ) = / ( \ , } ' V( a - 1 A ( a - 3; {,

J--1 ,«=l f

(IV.43);

(T + l ) a - i - l

using (IV.27). Based on (IV.39) and (IV.40) the reader can give the analogous formulae for P = - , (3 = 1 (hence a = 3, a = 2).

2. Second method This method only applies to the calculation of V2(g) since we use the formula V2 (g) = — , where a 2 and (j. are the variance and average of the Zipf function g, which can be calculated f using the Lotka function — as the weight function (cf. formula (11.20)):

» =l

C~ f(j)

J^dj

(IV.44)

So

V 2 (g) = ^ - r P m j 2 f(j)dj-l

(IV.45)

where n = —, the average number of items per source. We have by (11.34) and (11.35), if a > 1, a ^ 2 that

210


T=

I^ a ( p "" 1 )

(IV 46)

r - l )

(IV-47)

'

and that

A= ^ (

P

Also, if a * 3,

j;Pmj2f(J)dJ = XP"']grdj = ^ - ( p - - l )

(IV.48)

(IV.46), (IV.47) and (IV.48) in (IV.45) now gives

i8J

"(«-l)(a-3)-

v-(g)= ^ ^

2

(«-l)(a-3)

(p^-1)2

(pr-iKpr-i)^ 2 (pr_!)

and we have the task of proving that (IV.49) and (IV.43) are the same. Since we have Zipf s law, (11.91) implies that

,_a oc-1 Pm = - £ -

This and (11.34) or (IV.46) give

T+l=- ^ - =pr' a-1

(IV.50)


211

Formula (IV.50) is the link between (IV.49) and (IV.43). We leave it to the reader to consider the cases a = 2 and a = 3 .

3. Limiting case Forxm —> oo, formula (IV.38) gives, if y < —

lim V 2 (h) = ^ — l L - \ = jf— l-2y l-2y *»— V '

(IV.51)

This gives for V2(g) if p = y < — (implying a > 3):

limV 2 (g) = - ^ — = 1 TÔO W l-2p (a-l)(a-3)

(IV.52)

using again (IV.27). Note again that these formulae are in agreement with Corollary IV.3.2.1.5, as they should be. Formula (IV.51) also follows by direct calculation based on (IV.26) (here we also find the restriction y < — ).

IV.3.2.2.3 Calculation of Theil's measure We have to evaluate (IV. 19) with L(h)' as in (IV.36) (y *= 1) or as in (IV.37) (y = 1). We will limit ourselves to the general case y ^ 1, leaving the other calculation to the reader. The calculation is straightforward but tedious. We have

Th(h) = / o 1 L(h)'(y)ln(L(h)'(y))dy

J

° ( x ^ - l ) ( y ( x m - l ) + l)y

( x ^ - l ) ( y ( X i n - l ) + l)T]

212


_(1- Y )(x m -1)

(1- Y )(x m -1) r'

^-1 +

dy

Jo (y(Xm _ l) + 1y

C-l

(i-r)(x - i ) j . ( y ( X m _ 1 ) +

irin|(y(Xm_1)+ir|dy

m

, »?-' 1 p^_ifa) r Vnnzdz x^-1

Jl

z

x^ T -lJi

'

using the substitution z = y (xm — l) + 1 .

Now we use the following integral results which can readily be checked r dz _ z'~T

J^"TT^ rhiz , _ J

7"

Z

1

lnz

1

~-^ T 7^T (y-l)2

We then obtain

mM-h[fizî|+^-0^-0

(IV,3)

The measure Th(h) for xm = oo (restricted to y < 1, see (IV.26)) can be obtained in two ways: using (IV.26) and noting that L(h)'(y) = (l — y)y"y and reperforming the above calculation of Th(h), or, more simply, take lim Th(h) in (IV.53). Both ways give the X m —toe

formula (for y < 1)


Th(h) = ln(l-y) + - ^ -

213

(IV.54)

For Zipf s function (IV.22) (the most important case) this gives (for (3 < 1)

Th(g) = ln(l-(3) + J - .

In terms of Lotka's exponent a we have B = {

(IV.55)

(hence a > 2): a-lj

Th(g) = l n f ^ ) + - L - . l,ot —1J

(IV.56)

a-2

Note also that Th(g) increases in p and decreases in a as predicted by Corollary IV.3.2.1.5.

Note: Theil's measure is (as the other measures G and V2) calculated on the Lorenz curve, hence on normalized data in terms of abscissa and ordinate (see Figs. 1 and 2). It is different from the classical formula for the entropy of a distribution. To see this difference, let us first look at the discrete case. Theil's measure of the vector X = (x,,...,x N ) is given by (IV. 11), where the up are given by (IV.5). In the same notation, the entropy of the vector X is defined as (use log = In)

N

H = H(X) = - J ] a i l n a i

(IV.57)

i=i

Hence we have the relation

Th(X) + H(X) = lnN

or

(IV.58)

214

Power laws in the information production process: Lotkaian informetrics Th(X) = - H ( X ) + lnN

(IV.59)

We notice two differences between Th and H : although they are linearly related, the relation is decreasing. In other words, H increases if and only if Th decreases. Now Th is a good measure of concentration (inequality), hence H is a good measure of dispersion (i.e. of equality), used e.g. in biology to measure diversity (see Rousseau and Van Hecke (1999)). Further |Th| ^ H due to the fact that Th is calculated on the normalized Lorenz curve while H is not.

In the same way Th and H are different in the continuous case. We calculated already Th above. In the same way Lafouge and Michel (2001) calculate H (in fact, our proof of the Th formula was based on their proof) via the formula (f = Lotka function)

H(f) = -Jj°°f(j)ln(f(j))dj

(IV.60)

H(f) = - l n ( a - l ) + - i - + l

(IV.61)

For a > 1, they find

(note that H(f) decreases with a , in agreement with Corollary IV.3.2.1.5 since —H(f) is a good concentration measure). The result, apparently, was first stated by Yablonsky (1980). For H (g), the same formula (IV.61) applies but with a replaced by (5 > 1 (hence 1 < a < 2).

IV.3.3

A characterization of Price's law of concentration in terms of Lotka's law and of Zipf s law

In Theorem IV.3.2.1.1 we expressed the arithmetic fraction of the sources in [l,T + l] by numbers of the form

boiling down to an arithmetic fraction in [0,T]. For a geometric


215

fraction (needed in the formulation of Price's law) we cannot use the interval [0,T] since T e ,6 € [0,l] has the minimal value 1 (for 0 = 0). This, once more, shows the value of the use of the interval [l,T + l]: here a geometric fraction is perfectly expressed by the form (T + l) , Q £ [0,1]. This explains the used intervals in the following theory on Price's law.

The following proposition is a continuous extension of a result in Egghe and Rousseau (1986).

Proposition IV.3.3.1: Denote by G(r), for r e [l,T +1], the cumulative number of items produced by the sources in the interval [l,r]. Then the following assertions are equivalent:

(0 G(r) = Blnr

(IV.62)

r 6 [l,T +1], where B is a constant,

(ii)

The law of Price is valid: for every 9e]0,l[ (hence also for [0,l]), the top (T +1) sources produce a fraction 0 of the items.

Proof: (i)=>(ii)

Since we have the top (T + l) (0 G ]0,l[) sources, their cumulative production is given by (definition of G and since (T + i f € [l,T +1]):

G((T + l) 9 ) = Bln((T + l) 9 )

G((T + l) 8 ) = eG(T + l)

216


But G(T + l) denotes the total number of items, hence the top (T + l) sources produce a fraction 8 of the items.

(iiWi)

Let r e [l,T +1] be arbitrary. Hence there exists Q € [0,1] such that r = (T +1) 6 . By (ii) and the definition of G we have that

G ((T

+ l) 9 ) = eG(T + l)

for 0 e ]0,l[. This is also true for 6 = 1 and for 9 = 0 (since we work with source densities). But r = (T +1) implies lnr = 61n(T +1). Hence

G (Vr;) =

tar ,G(T + 1) ln(T + l) l '

G(r) = Blnr

G(T + 1) where B = —-. '-, a constant. ln(T + l)

Corollary IV.3.3.2 (Egghe (2004c)): We have the equivalence of the following statements

(i)

Price's law is valid

(ii)

Zipf s law (IV.22) is valid for (3 = 1.

(iii)

Lotka's law (IV.21) is valid for a = 2 and

Pm

=C

Proof: The equivalence of (ii) and (iii) was proved in Subsection II.4.2.

(IV.63)


217

As to the equivalence of (i) and (ii) it suffices, by Proposition IV.3.3.1, to show that Zipfs law for (3 = 1 is equivalent with G(r) = Blnr, r € [l,T + 1 where B is a constant. In other words we have to show that

E

I \

r £ [0,T] is equivalent with

G(r) = Bln(l + r)

r £ [0,T]. This follows from formula (II.9) in Subsection II. 1.2 or from Theorem II.2.2.1 and formula (11.48).

We have the following (for historic reasons - see further) remarkable corollary:

Corollary IV.3.3.3 (Allison, Price, Griffith, Moravcsik and Stewart (1976) for Price's square root law and Egghe (2004c) for the general case): If a = 2 then Price's law is equivalent with Lotka's law for which the following relation is valid:

Pm

= C

(IV.64)

Proof: This follows readily from the previous corollary.

The result in Corollary IV.3.3.3 was found in Allison, Price, Griffith, Moravcsik and Stewart (1976) after a long approximative calculation (and limited to Price's law for 9 = — , i.e. Price's square root law; that the result is valid for the general law of Price was first proved, exactly, in Egghe (2004c)). This paper apparently (see editor's note) grew out of lengthy and frequently heated correspondence between these authors on the validity of Price's square root

218


law in case of Lotka's law. We hereby show that a long debate on this issue is not necessary since even a more general result can be proved in an exact way.

Note also that the results in Proposition IV.3.3.1 and Corollary IV.3.3.2 are exact formulations of a "feeling" of De Solla Price (in De Solla Price (1963)) that Lotka's law, Price's law and Zipfs law (there called Pareto's law) describe "something like the approximate law of Fechner or Weber in experimental psychology, wherein the true measure of the response is taken not by the magnitude of the stimulus but by its logarithm; we must have equal intervals of effort corresponding to equal ratios of numbers of publications" (p.50).

We refer the reader to Glanzel and Schubert (1985) for a discrete characterization of Price's (square root, i.e. 0 = —) law. Further discrete calculations on Lotka's law in the connection of Price's law can be found in Egghe (1987a). Practical investigations on the validity of Price's law can be found in Berg and Wagner-Dobler (1996), Nicholls (1988) and Gupta, Sharma and Kumar (1998). We also give the reference Bensman (1982) for general examples of concentration in citation analysis and in library use.

IV.4

CONCENTRATION

THEORY

OF

LINEAR

THREE-

DIMENSIONAL INFORMETRICS Three-dimensional informetrics was described in Chapter III, which was almost entirely devoted to linear three-dimensional informetrics. The composition of the two IPPs was proved to be a situation of positive reinforcement i.e. where the rank-frequency function of the first IPP, denoted by g,, was "reinforced" as g = cp°g, with 9 a strictly increasing function such that cp(x) > x and where g is the rank-frequency function of the composed IPP. In positive reinforcement we hence keep the source set and we reinforce the production of these sources. Also in Chapter III, one studies Type/Token-Taken (TTT) informetrics where the use (in the second IPP) of the items (in the first IPP) was studied. Both cases yield a new IPP of which we have proved informetric properties in Chapter III.


219

It is, hence, natural to investigate the concentration properties of the positively reinforced IPP as well as the concentration properties of TTT informetrics. This will be the topic of this section.

IV.4.1 The concentration of positively reinforced IPPs

The reader is requested to re-read Subsection III. 1.3.1. To put it simply: the rank-frequency function g[ of the first IPP is composed with a strictly increasing function cp given by (III.5) or (III.6). Let us denote by L(g,) the Lorenz curve of gj and by L(g) = L ^ g ^ the Lorenz curve of g = L (g,) if — ^ is increasing

(ii)

L ((p°g,) = L (g,) if ^-L

is constant

X

(iii)

L (cpogl) < L (g,) if W

is decreasing

Strict inequalities apply in (i) and (iii) when (ii) is not the case.

Proof: Note that Ti equals the total number of sources in the first as well as in the composed IPP (only the productivity is changed, by cp). By (IV. 13) we have:

L(cp°gl)(y) = V ,

and

^ '

dV.65)

220

Power laws in the information production process: Lotkaian informetrics F T l + 1 I 0 on ]0,yJ 81 (y)

and A ' ( y ) < 0 on ]y o ,l[. Hence, since A(0) = A(l) = O we have that A(y)>0 on [0,l], hence

L(< P °g 1 )(y)>L(g 1 )(y)

for all y e [0,l], proving (i). (iii) is proved in the same way and, by (IV.74) and (IV.76) (ii) is trivial. By the above proof, it is clear that strict inequalities apply in cases (i) and (iii) when (ii) is not the case.

The above theorem (in case —^—^ increases), restricted to the discrete case, was proved in x Rousseau (1992a) but Burrell (1992a) pointed out the existence of the general result of Fellman(1976). This theorem gives the complete situation of positive reinforcement. Based on Theorem IV.4.1.1 we do not always have a more concentrated situation after positive reinforcement. Indeed, it is clear that, although cp implies an increase in the number of items per source of


223

the first IPP, this does not always imply an increase in concentration. Indeed, examples abound: one can add items to each source so that they all have equal production, in which case we even have equality, hence the lowest concentration that is possible (and L((p°g,)(y) = y on [0,l]). Another, more realistic case is given by adding to all productions, the same amount of items so that we are in situation (vi) in Section IV. 1 (the principle of nominal increase), hence a decrease of concentration, hence an L((p°g1) which is lower than L(g,)-

The following corollary trivially follows from Theorem IV.4.1.1.

Corollary IV.4.1.2: Let 9 be a power law of the type cp(x) = bxa with a, b positive parameters. Then L ( c p o g l ) > L ( g l ) i f f a > l , L( 0 and

238


Thus the graph of ^" (F) in function of s (see Fig.V.2) shows that there is exactly one critical value of s at which ^

(F) jumps from +00 to 0. This critical value of s is called the

Hausdorff-Besicovitch dimension and is denoted as s = dimH (F).

Fig. V.2

Graph of

Power Laws in the Information Production Process: Lotkaian Informetrics (Library and Information Science)

Text Information Retrieval Systems, Third Edition (Library and Information Science) (Library and Information Science) (Library and Information Science)

Career Opportunities in Library and Information Science

Opportunities in Library and Information Science

Communicating Research (Library and Information Science Series) (Library and Information Science)

Information Science

Library and Information Center Management (Library and Information Science Text Series)

Marketing Library and Information Services

Global Librarianship (Books in Library and Information Science)

Introduction to the Library and Information Professions

Informed Consent: Information Production and Ideology

Geographic Information: Value, Pricing, Production, and Consumption

Finding Information in Science, Technology and Medicine

Multidimensional geographic information science

The Power Laws: The Science of Success

The Power Laws: The Science of Success

Looking for Information: A Survey of Research on Information Seeking, Needs, and Behavior (Library and Information Science)

Looking for Information, Second Edition: A Survey of Research on Information Seeking, Needs, and Behavior (Library and Information Science)

Introduction to Technical Services (Library and Information Science Text Series)

Library and Information Science Annual: 1999 Volume 7

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