Dynamic evolution in societal networks

Abstract

A societal network is defined to be a finite directed graph in which individuals are represented by nodes, and relations between individuals by labelled arcs. Each individual undergoes state transitions at discrete instants of time, so that the societal network may be thought of as a deterministic dynamic process. It is shown that the individuals of such a network may be divided into equivalance classes, so that the original network may be represented by a reduced network containing one node for each equivalence class. Similar results are obtained for a more general type of partition called a “class structure.” The application of these concepts is illustrated in the context of “balanced” networks, which have either positive or negative relations between nodes. The long‐run behavior of societal networks is then examined, and it is shown that any network will eventually reach either a stable state or a periodic pattern of state transitions. In the case of a probabilistic transition rule, it is proved that the long‐run evolutionary pattern is independent of the initial state of the network. Finally, some potential areas of future work are suggested.