Effects of mixing in threshold models of social behavior

TL;DR

This paper studies how mixing and network effects influence the dynamics of social behavior models, revealing how information flow impacts the persistence or change of social norms.

Contribution

It extends the Granovetter model by analyzing the effects of mixing and network structure on social behavior dynamics, combining analytical and numerical methods.

Findings

Finite mixing increases the likelihood of shifting to the ground state.

Network effects can be approximated by finite-neighborhood effects.

Dynamics tend to converge to a manifold determining possible equilibria.

Abstract

We consider the dynamics of an extension of the influential Granovetter model of social behavior, where individuals are affected by their personal preferences and observation of the neighbors' behavior. Individuals are arranged in a network (usually, the square lattice) and each has a state and a fixed threshold for behavior changes. We simulate the system asynchronously either by picking a random individual and either update its state or exchange it with another randomly chosen individual (mixing). We describe the dynamics analytically in the fast-mixing limit by using the mean-field approximation and investigate it mainly numerically in case of a finite mixing. We show that the dynamics converge to a manifold in state space, which determines the possible equilibria, and show how to estimate the projection of manifold by using simulated trajectories, emitted from different initial points. We show that the effects of considering the network can be decomposed into finite-neighborhood effects, and finite-mixing-rate effects, which have qualitatively similar effects. Both of these effects increase the tendency of the system to move from a less-desired equilibrium to the "ground state". Our findings can be used to probe shifts in behavioral norms and have implications for the role of information flow in determining when social norms that have become unpopular (such as foot binding or female genital cutting) persist or vanish.