Long-Term Opinion Distributions of an Opinion Formation Model with Averaging Behavior

Abstract

Sociophysics utilizes mathematical tools from physics to study social phenomena. In particular, interactions between individuals can impact emerging trends in the opinions of a population as a whole. This research focuses on the long-term opinion distributions of a population of individuals who can influence one another through pairwise interactions, where one individual modifies their opinion to be more in line with their neighbor. Interactions are governed by a social network, where friends on the network interact while strangers do not. We introduce a new model for opinion formation with averaging behavior where the opinion of an individual at any time t is an integer between −k and k. For example, an opinion of k could indicate a heavily republican opinion and −k a heavily democratic opinion. As the process evolves in time, interactions between neighbors x and y in the social network result in person x updating her opinion to be one step closer to the opinion of person y. We first consider the scenario where everyone in the social network is friends with everyone else. For k = 1 we compute explicitly the long-term opinion distribution via differential equations. For values of k > 1, we solve for the long-term distribution numerically using Runge-Kutta. We then compute the long-term opinion distribution via simulation, in the more general social network scenario.