Multi-body Interactions and Non-Linear Consensus Dynamics on Networked Systems

TL;DR

This paper explores how multi-body, non-linear interactions on hypergraphs influence consensus dynamics, revealing effects like state shifts and cluster dominance that differ from traditional pairwise models.

Contribution

It introduces a non-linear model for three-body consensus on hypergraphs, highlighting effects beyond pairwise interactions and analyzing their impact on system dynamics.

Findings

Multi-body effects require non-linearity to manifest.

Dynamics can cause shifts away from the average state.

Multi-body interactions can lead to asymmetric cluster dominance.

Abstract

Multi-body interactions can reveal higher-order dynamical effects that are not captured by traditional two-body network models. In this work, we derive and analyse models for consensus dynamics on hypergraphs, where nodes interact in groups rather than in pairs. Our work reveals that multi-body dynamical effects that go beyond rescaled pairwise interactions can only appear if the interaction function is non-linear, regardless of the underlying multi-body structure. As a practical application, we introduce a specific non-linear function to model three-body consensus, which incorporates reinforcing group effects such as peer pressure. Unlike consensus processes on networks, we find that the resulting dynamics can cause shifts away from the average system state. The nature of these shifts depends on a complex interplay between the distribution of the initial states, the underlying structure and the form of the interaction function. By considering modular hypergraphs, we discover state-dependent, asymmetric dynamics between polarised clusters where multi-body interactions make one cluster dominate the other.