Stochastic Dynamics on Hypergraphs and the Spatial Majority Rule Model

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 × · · · × 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.

Opinion Dynamics: Models, Extensions and External Effects

“…The majority rule model has been analytically studied on hypergraphs, with a version entitled 'spatial majority rule model' [50]. Hyperedges consisting of n vertices were used to define social groups.…”

SIS Epidemic Propagation on Hypergraphs

“…, c k }, therefore S k−1 C k−1 = γkS k , proving the first part of the statement. The second part follows from the first one by using equation (12).…”

Deterministic Versus Stochastic Consensus Dynamics on Graphs

“…Such models are often posed in an agent based framework where the potential for agents to interact is encoded in a graph; models of opinion formation are especially amenable to such a framework and a considerable amount of research has been done in this context [11,19,27,29,36]. Many of the dynamics studied are deterministic in nature [5,6,19,34], however the random nature of information exchange among humans strongly motivates the study of similar models in a stochastic setting [1,14,26]. In this paper we investigate two models of opinion formation, one stochastic and one deterministic with special attention paid to the emergence of consensus -when the opinions of all agents agree.…”

“…The stochastic nature of these models causes their analysis to require a different set of mathematical tools, most frequently from the theory of Markov chains and martingales. Similar to the deterministic case, a sufficient degree of interaction among agents is often necessary for a consensus to emerge [2,25,26].…”