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The Probability of a given 1-Choice Structure

Published online by Cambridge University Press:  01 January 2025

Frank Harary  and
Ron Read
Frank Harary
Affiliation:
University of Michigan
Ron Read
Affiliation:
University of the West Indies
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Abstract

A 1-choice structure arises whenever each person in a group chooses exactly one other person according to some criterion. Our purpose is to study the situation in which these choices are made at random. As noted in Harary, Norman and Cartwright [2], such a structure can be represented by a directed graph in which the points represent people and the directed lines their choices. We first describe the shape of such a 1-choice structure, and define its symmetry number. With the help of these properties we are then able to develop and prove a formula which gives a probability of obtaining a given structure in the random choice situation. In order to supply data for these results, we include in the Appendix the diagrams of all 1-choice structures with at most 7 points and the probability of each.

Type
Original Paper
Information
Psychometrika , Volume 31 , Issue 2 , June 1966 , pp. 271 - 278
Copyright
Copyright © 1966 Psychometric Society

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Footnotes

*

The preparation of this article was supported by Grant MH 10834 from the National Institute of Mental Health and Grant 1026-66 from the U. S. Air Force Office of Scientific Research.

References

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Read, R. C. A note on the number of functional digraphs. Math. Annalen, 1961, 143, 109–110.Google Scholar