In this paper, we consider a disjoint decomposition, at three levels, of the total sample space for $n$-person, one-dimensional sociometric investigations. This results in a structure particularly suited to determination of the probability distributions of a large class of sociometric variables. Systematic methods for obtaining these distributions are presented and illustrated by two examples; while the first is trivial, the second produces a previously unknown result. It should be remarked that the methods developed here have application in the theory of communication networks and, indeed, in the study of any network situations which may be represented by either of the two models employed in the paper.
"Probability Distributions of Random Variables Associated with a Structure of the Sample Space of Sociometric Investigations." Ann. Math. Statist. 28 (2) 442 - 448, June, 1957. https://doi.org/10.1214/aoms/1177706972