Power Laws in the Information Production Process : Lotkaian Informetrics

by Egghe

• ISBN13: 9780120887538
• ISBN10: 0120887533
• Format: Hardcover
• Copyright: 2005-03-23
• Publisher: Elsevier Science & Technology

Summary

This book describes informetric results from the point of view of Lotkaian size-frequency functions, i.e. functions that are decreasing power laws. Explanations and examples of this model are given showing that it is the most important regularity amongst other possible models. This theory is then developed in the framework of IPPs (Information Production Processes) hereby also indicating its relation with e.g. the law of Zipf. Applications are given in the following fields: three-dimensional informetrics (positive reinforcement and Type/Token-Taken informetrics), concentration theory (including the description of Lorenz curves and concentration measures in Lotkaian informetrics), fractal complexity theory (Lotkaian informetrics as self-similar fractals), Lotkaian informetrics in which items can have multiple sources (where fractional size-frequency functions are constructed), the theory of first-citation distributions and the N-fold Cartesian product of IPPs (describing frequency functions for N-grams and N-word phrases). In the Appendix, methods are given to determine the parameters in the law of Lotka, based on a set of discrete data. The book explains numerous informetric regularities, only based on a decreasing power law as size-frequency function, i.e. Lotka's law. It revives the historical formulation of Alfred Lotka of 1926 and shows the power of this power law, both in classical aspects of informetrics (libraries, bibliographies) as well as in "new" applications such as social networks (citation or collaboration networks and the Internet).

Table of Contents

Preface	vii
Table of contents	xi
Introduction	1	(6)
Lotkaian Informetrics: An Introduction	7	(94)
Informetrics	7	(7)
What is Lotkaian informetrics?	14	(11)
The law of Lotka	14	(5)
Other laws that are valid in Lotkaian informetrics	19	(6)
Why Lotkaian informetrics?	25	(60)
Elementary general observations	26	(1)
The scale-free property of the size-frequency function f	27	(5)
Power functions versus exponential functions for the size-frequency function f	32	(2)
Proof of Lotka's law based on exponential growth or based on exponential obsolescence	34	(1)
Proof of Lotka's law based on exponential growth: the Naranan model	34	(6)
Proof of Lotka's law based on exponential obsolescence: solution of a problem of Buckland	40	(2)
Derivation of Mandelbrot's law for random texts	42	(3)
``Success Breeds Success''	45	(1)
The urn model	46	(3)
General definition of SBS in general IPPs	49	(3)
Approximate solutions of the general SBS	52	(3)
Exact results on the general SBS and explanation of its real nature	55	(10)
Entropy aspects	65	(1)
Entropy: definition and properties	66	(4)
The Principle of Least Effort (PLE) and its relation with the law of Lotka	70	(6)
The Maximum Entropy Principle (MEP)	76	(2)
The exact relation between (PLE) and (MEP)	78	(7)
Practical examples of Lotkaian informetrics	85	(16)
Important remark	85	(1)
Lotka's law in the informetrics and linguistics literature	86	(1)
Lotka's law in networks	87	(3)
Lotka's law and the number of authors per paper	90	(2)
Time dependence and Lotka's law	92	(2)
Miscellaneous examples of Lotkaian informetrics	94	(4)
Observations of the scale-free property of the size-frequency function f	98	(3)
Basic Theory of Lotkaian Informetrics	101	(56)
General informetrics theory	101	(13)
Generalized bibliographies: Information Production Processes (IPPs)	101	(3)
General informetric functions in an IPP	104	(6)
General existence theory of the size-frequency function	110	(4)
Theory of Lotkaian informetrics	114	(30)
Lotkaian function existence theory	114	(1)
The case pm = ∞	114	(2)
The general case pm < ∞	116	(5)
The informetric functions that are equivalent with a Lotkaian size-frequency function f	121	(23)
Extension of the general informetrics theory: the dual size-frequency function h	144	(6)
The place of the law of Zipf in Lotkaian informetrics	150	(7)
Definition and existence	150	(2)
Functions that are equivalent with Zipf's law	152	(5)
Three-dimensional Lotkaian Informetrics	157	(30)
Three-dimensional informetrics	157	(18)
The case of two source sets and one item set	158	(1)
The case of one source set and two item sets	159	(2)
The third case: linear three-dimensional informetrics	161	(2)
Positive reinforcement	163	(5)
Type/Token-Taken informetrics	168	(4)
General notes	172	(3)
Linear three-dimensional Lotkaian informetrics	175	(12)
Positive reinforcement in Lotkaian informetrics	175	(2)
Lotkaian Type/Token-Taken informetrics	177	(10)
Lotkaian Concentration Theory	187	(44)
Introduction	187	(5)
Discrete concentration theory	192	(4)
Continuous concentration theory	196	(22)
General theory	196	(3)
Lotkaian continuous concentration theory	199	(1)
Lorenz curves for power laws	199	(6)
Concentration measures for power laws	205	(9)
A characterization of Price's law of concentration in terms of Lotka's law and of Zipf's law	214	(4)
Concentration theory of linear three-dimensional informetrics	218	(13)
The concentration of positively reinforced IPPs	219	(7)
Concentration properties of Type/Token-Taken informetrics	226	(5)
Lotkaian Fractal Complexity Theory	231	(16)
Introduction	231	(1)
Elements of fractal theory	232	(10)
Fractal aspects of a line segment, a rectangle and a parallelepiped	233	(1)
The triadic von Koch curve and its fractal properties. Extension to general self-similar fractals	234	(2)
Two general ways of expressing fractal dimensions	236	(1)
The Hausdorff-Besicovitch dimension	236	(3)
The box-counting dimension	239	(3)
Interpretation of Lotkaian IPPs as self-similar fractals	242	(5)
Lotkaian Informetrics of Systems in which Items can have Multiple Sources	247	(48)
Introduction	247	(6)
Crediting systems and counting procedures for sources and ``super sources'' in IPPs where items can have multiple sources	253	(23)
Overview of crediting systems for sources	254	(1)
First or senior author count	254	(1)
Total author count	254	(1)
Fractional author count	255	(1)
Proportional author count	255	(1)
Pure geometric author count	255	(1)
Noblesse Oblige	256	(1)
Crediting systems for super sources	256	(1)
Counting procedures for super sources in an IPP	256	(1)
Total counting	257	(1)
Fractional counting	258	(1)
Proportional counting	258	(3)
Inequalities between QT (c) and QF (c) and consequences for the comparison of QT (c), QF (c) and Qp (c)	261	(5)
Solutions to the anomalies	266	(1)
Partial solutions	267	(2)
Complete solution to the encountered anomalies	269	(1)
Conditional expectation results on QT (c), QF (c) and Qp (c)	270	(6)
Construction of fractional size-frequency functions based on two dual Lotka laws	276	(19)
Introduction	276	(2)
A continuous attempt: z ε R+	278	(4)
A rational attempt: q ε Q+	282	(13)
Further Applications in Lotkaian Informetrics	295	(70)
Introduction	295	(2)
Explaining ``regularities''	297	(3)
The arcs at the end of a Leimkuhler curve	297	(1)
A ``type/token-identity'' of Chen and Leimkuhler	298	(2)
Probabilistic explanation of the relationship between citation age and journal productivity	300	(4)
General and Lotkaian theory of the distribution of author ranks in multi-authored papers	304	(9)
General theory	304	(4)
Modelling the author rank distribution using seeds	308	(2)
Finding a seed based on alphabetical ranking of authors	310	(3)
The first-citation distribution in Lotkaian informetrics	313	(13)
Introduction	313	(4)
Derivation of the model	317	(3)
Testing of the model	320	(1)
First example: Motylev (1981) data	320	(2)
Second example: JACS to JACS data of Rousseau	322	(1)
Extensions of the first-citation model	323	(3)
Zipfian theory of N-grams and of N-word phrases: the Cartesian product of IPPs	326	(39)
N-grams and N-word phrases	326	(3)
Extension of the argument of Mandelbrot to 2-word phrases	329	(4)
The rank-frequency function of N-grams and N-word phrases based on Zipf's law for N = 1	333	(14)
The size-frequency function of N-grams and N-word phrases derived from Subsection VII.6.3	347	(5)
Type/Token averages μn and Type/Token-Taken averages μn for N-grams and N-word phrases	352	(13)
Appendix	365	(32)
Appendix I	365	(5)
Appendix II	370	(2)
Appendix III Statistical determination of the parameters in the law of Lotka	372	(25)
A.III.1 Statement of the problem	372	(1)
A.III.2 The problem of incomplete data (samples) and Lotkaian informetrics	373	(5)
A.III.3 The difference between the continuous Lotka function and the discrete Lotka function	378	(8)
A.III.4 Statistical determination of the parameters K, a, nmax in the discrete Lotka function K/na, n = 1,..., nmax	386	(1)
A.III.4.1 Quick and Dirty methods	387	(1)
A.III.4.2 Linear Least Squares method	388	(2)
A.III.4.3 Maximum Likelihood Estimating method	390	(3)
A.III.5 General remarks	393	(1)
A.III.5.1 Fitting Zipf's function	393	(1)
A.III.5.2 The estimation of pm and nmax	394	(1)
A.III.5.3 Fitting derived functions such as Price's law	394	(1)
A.III.5.4 Goodness-of-fit tests	395	(2)
Bibliography	397	(26)
Subject Index	423