An urn model for opinion propagation on networks

We consider a coupled Polya’s urn scheme for social dynamics on networks. Agents hold continuum-valued opinions on a two-state issue and randomly converse with their neighbors on a graph, agreeing on one of the two states. The probability of agreeing on a given state is a simple function of both of agents’ opinions, with higher importance given to agents who have participated in more conversations. Opinions are then updated based on the results of the conversation. We show that this system is governed by a discrete version of the stochastic heat equation, and prove that the system reaches a consensus of opinion.