Phase transitions and stability of dynamical processes on hypergraphs

TL;DR

This paper develops a theoretical framework to analyze the stability of dynamical processes on hypergraphs, revealing how higher-order interactions influence processes like social contagion and diffusion.

Contribution

It introduces the first general framework for studying dynamical processes on hypergraphs, linking stability to hypergraph structure and interaction order.

Findings

Stability near fixed points depends on the graph-projection of the hypergraph.

In social contagion, pairwise interactions determine stability.

For diffusion, the interaction order affects stability differently.

Abstract

Hypergraphs naturally represent higher-order interactions, which persistently appear from social interactions to neural networks and other natural systems. Although their importance is well recognized, a theoretical framework to describe general dynamical processes on hypergraphs is not available yet. In this paper, we bridge this gap and derive expressions for the stability of dynamical systems defined on an arbitrary hypergraph. The framework allows us to reveal that, near the fixed point, the relevant structure is the graph-projection of the hypergraph and that it is possible to identify the role of each structural order for a given process. We also analytically solve two dynamics of general interest, namely, social contagion and diffusion processes, and show that the stability conditions can be decoupled in structural and dynamical components. Our results show that in social contagion processes, only pairwise interactions play a role in the stability of the absorbing state, while for the diffusion dynamics, the order of the interactions plays a differential role. Ours is the first attempt to provide a general framework for further exploration of dynamical processes on hypergraphs.