Dynamical influence processes on networks: General theory and applications to social contagion

TL;DR

This paper develops a general mean field theory for binary state dynamics on networks, capturing how local responses and network structure influence global behaviors like social contagion, from steady states to chaos.

Contribution

It introduces a unified framework for analyzing stochastic and deterministic network dynamics, including a novel model of social contagion with imitation and non-conformity.

Findings

Mean field theory accurately predicts network dynamics.

Dynamics converge in large, dense networks.

Model captures complex social phenomena like trends and fashions.

Abstract

We study binary state dynamics on a network where each node acts in response to the average state of its neighborhood. Allowing varying amounts of stochasticity in both the network and node responses, we find different outcomes in random and deterministic versions of the model. In the limit of a large, dense network, however, we show that these dynamics coincide. We construct a general mean field theory for random networks and show this predicts that the dynamics on the network are a smoothed version of the average response function dynamics. Thus, the behavior of the system can range from steady state to chaotic depending on the response functions, network connectivity, and update synchronicity. As a specific example, we model the competing tendencies of imitation and non-conformity by incorporating an off-threshold into standard threshold models of social contagion. In this way we attempt to capture important aspects of fashions and societal trends. We compare our theory to extensive simulations of this "limited imitation contagion" model on Poisson random graphs, finding agreement between the mean-field theory and stochastic simulations.