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On time changes in a digraph

Published online by Cambridge University Press:  14 July 2016

T. N. Bhargava
T. N. Bhargava*
Affiliation:
Kent State University
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Extract

The object of this paper is to present a probabilistic model for analyzing changes through time in a binary dyadic relation on a finite set of points. The total relation on the set takes the form of an aggregate of directed binary dyadic relations between ordered pairs of points belonging to the set; equivalently, the total relation on the set can be represented by means of a digraph or an incidence matrix isomorphic with the total relation. Such a relation, changing in its structure as time proceeds, is a reasonable mathematical model, for example, for the evolution of inter-relationships of the members of a social or any other group. A group of this kind is organized for a specific activity involving some sort of “communication” from one member to the other, and may be observed at successive discrete points in time generating statistics on the evolutionary process. (For a detailed treatment, see [3].) As a matter of fact, under suitable assumptions, the model presented here has potentialities for application in those situations which can be represented mathematically in terms of a finite set of points and an all-or-none relationship between ordered pairs of these points. Some of the other examples are communication networks, ecology, animal sociology, and management sciences (see [5]).

Type
Research Papers
Information
Journal of Applied Probability , Volume 2 , Issue 1 , June 1965 , pp. 101 - 118
Copyright
Copyright © Sheffield: Applied Probability Trust 

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References

[1] Anderson, T. W. and Goodman, L. A. (1957) Statistical inference about Markov chains Ann. Math. Statist. 28, 89–110.Google Scholar
[2] Bhargava, T. N. (1962) A stochastic model for a binary dyadic relation with application to group dynamics. Ph. D. Thesis, Michigan State University.Google Scholar
[3] Bhargava, T. N. On treatment of group dynamics as a stochastic process (to appear).Google Scholar
[4] Bhargava, T. N. and Katz, Leo (1963) A stochastic model for a binary dyadic relation with applications to social and biological sciences. Presented at the 34 th Session of the International Statistical Institute, Ottawa.Google Scholar
[5] Feller, W. (1950) An Introduction to Probability Theory and its Applications. John Wiley, New York.Google Scholar
[6] Fréchet, M. (1940 and 1943) Les probabilités associées à un système d'évènements compatibles et dépendants. Actualités Scientifiques et Industrielles 859 and 942.Google Scholar
[7] Katz, L. and Wilson, T. R. (1956) The variance of the number of mutual choices in sociometry. Psychometrika 21.Google Scholar
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