Stochastic dynamics on hypergraphs and the spatial majority rule model

TL;DR

This paper introduces a hypergraph-based framework for modeling stochastic spatial systems, applying it to a majority rule model where hyperedges influence opinions, revealing dimension-dependent consensus and clustering behaviors.

Contribution

It develops a novel hypergraph framework for spatial stochastic processes and analyzes a majority rule model with simultaneous hyperedge updates, highlighting new opinion dynamics phenomena.

Findings

Opinion 1 wins when hyperedges are even-sized in 2D.

System clusters when hyperedges are odd-sized.

Full proof of 1D case; analysis for 2D hyperedges of size 2 and 3.

Abstract

This article starts by introducing a new theoretical framework to model spatial systems which is obtained from the framework of interacting particle systems by replacing the traditional graphical structure that defines the network of interactions with a structure of hypergraph. This new perspective is more appropriate to define stochastic spatial processes in which large blocks of vertices may flip simultaneously, which is then applied to define a spatial version of the Galam's majority rule model. In our spatial model, each vertex of the lattice has one of two possible competing opinions, say opinion 0 and opinion 1, as in the popular voter model. Hyperedges are updated at rate one, which results in all the vertices in the hyperedge changing simultaneously their opinion to the majority opinion of the hyperedge. In the case of a tie in hyperedges with even size, a bias is introduced in favor of type 1, which is motivated by the principle of social inertia. Our analytical results along with simulations and heuristic arguments suggest that, in any spatial dimensions and when the set of hyperedges consists of the collection of all n x ... x n blocks of the lattice, opinion 1 wins when n is even while the system clusters when n is odd, which contrasts with results about the voter model in high dimensions for which opinions coexist. This is fully proved in one dimension while the rest of our analysis focuses on the cases when n = 2 and n = 3 in two dimensions.