A general theory of bibliometric and other cumulative advantage processes

A Cumulative Advantage Distribution is proposed which models statistically the situation in which success breeds success. I t differs from the Negative Binomial Die tribution in that lack of success, being a non-event, is not punished by increased chance of failure. I t is shown that such a stochastic law is governed by the Beta Function, containing only one free parameter, and this is approximated by a skew or hyperbolic distribution of the type that is widespread in bibliometrics and diverse social science phenomena. In particular, this is shown to 0 Introduction It is common in bibliometric matters and in many diverse social phenomena, that success seems to breed success. A paper which has been cited many times is more likely to be cited again than one which has been little cited. An author of many papers is more likely to publish again than one who has been less prolific. A journal which has been frequently consulted for some purpose is more likely to be turned to again than one of previously infrequent use. Words become common or remain rare. A millionaire gets extra income faster and easier than a beggar. In statistics, such a process is commonly described by a skew or hyperbolic distribution function of the type that has been characterized by Simon (I) and correctly shown by him to be given by the Beta Function (this is not the " Beta Density " which is discussed by Feller (2) 11:49 and includes a dependent variable) rather than the T h i s researckwas funded by NSF grant SOC73-05428. be an appropriate underlying probabilistic theory for the Bradford Law, the Lotka Law, the Pareto and Zipf Distributions , and for all the empirical results of citation frequency analysis. As side results one may derive also the obsolescence factor for literature use. The Beta Function is peculiarly elegant for these manifold purposes because it yields both the actual and the cumulative distributions in simple form, and contains a limiting case of an inverse square law to which many empirical distributions conform. more commonly used " contagious " distributions, for example the negative binomial (3. 4 3 or its limiting form, Fisher's logarithmic series distribution. Although the relation between such distributions, the stochastic processes which lead to them and the Urn models from which they may be derived, seem well known and go back more than 50 years to Yule's …