10
A CRITERION FOR UNANIMITY IN FRENCH'S THEORY OF SOCIAL POWER

FRANK HARARY

The purpose of this paper is to contribute to the theoretical investigation initiated by French (3) by providing a necessary and sufficient condition for the attainment of ultimate unanimity of opinions in a power structure. In addition the isomorphism is demonstrated between this theory and the theory of higher transition probabilities in Markov chains, as developed in the book by Feller (2). By exploiting this isomorphism the known theorems on Markov chains are translated into their corresponding theorems on social power. One of the results obtained in this manner is that every power structure regardless of initial opinion distribution necessarily converges to a stable distribution of ultimate opinions.

In the processes of deriving this criterion for unanimity, we employ several concepts of independent interest, including an "automorphic group" and a "power subgroup." We also propose a generalization of French's model, and find that the criterion for unanimity developed in French's model still remains valid in the more general theory.

French conceives of interpersonal power as the ability of one person to influence another by virtue of some more or less enduring basis of power (expertness, ability to punish, etc.) Variations in the strength of the basis of power effect variations in influence if and only if power is exerted, i.e., person A communicates an influence attempt to B, a person over whom he has power. Such a communication sets up a force field on B to agree with A's opinion, but this force field meets an opposing one corresponding to B's tendency to maintain his initial opinion unchanged. In the case of opinions measured along a cardinal scale, both these force fields are assumed to have linear gradients: the more A's opinion differs from B's the stronger force he induces on B; and the greater the change demanded of B, the stronger his opposition to the change.1 It is assumed that B resolves this conflict by changing his

____________________
1
French (3) originally called this "resistance" but more recently he distinguishes this "opposition" (based on own forces) from "resistance" (defined as forces induced by the act of A). The more recent usage is adhered to in this chapter.

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