Explaining homophily without social selection: The role of transitivity in the formation of homophilic ties

Abstract

This paper investigates if an assortative social network may further amplify its assortativity following two distinct interventions: linking a random unconnected pair in the neighborhood of a vertex chosen by chance (N-protocol) or connecting a randomly selected open triplet (T-protocol). Under a series of assumptions, we derive a closed-form expression that links the expected change in assortativity caused by N-protocol and the current assortativity score. For the binary nodal characteristic or some other settings, this expression turns out to be a simple quadratic dependency, which indicates that networks with assortative mixing should become less assortative following N-protocol. Next, we provide a sufficient condition that N- and T-protocols will cause the same assortativity change. Using numerical experiments with synthetic and empirical networks with various assortativity rates, we demonstrate that our theoretical estimations tend to be quite accurate for topologies with moderate assortativity. However, for topologies with relatively high assortativity levels, the factual values of the assortativity shift tend to be greater than theoretical ones, but still negative. Empirical topologies are also characterized by relatively large divergences between N- and T-protocols, with the latter typically providing a higher assortativity decrease. We carefully explain these findings by analyzing local homophily patterns. We also consider a modified version of T-protocol that connects individuals having no less than a predefined number of common peers. For a corpus of empirical networks, we managed to characterize the threshold values of assortativity and the number of common friends above which the assortativity rate will increase following this intervention.