Dynamical systems on Hypergraphs

TL;DR

This paper develops a framework for analyzing dynamical systems on hypergraphs, capturing complex multi-body interactions, and studies phenomena like Turing patterns and synchronization in such systems.

Contribution

It introduces a combinatorial Laplacian for hypergraphs and adapts the Master Stability Function formalism to analyze dynamics on hypergraphs.

Findings

Hypergraph Laplacian formalism is effective for modeling multi-body interactions.

Turing patterns emerge in hypergraph-based dynamical systems.

Synchronization behavior is influenced by higher-order interactions.

Abstract

Networks are a widely used and efficient paradigm to model real-world systems where basic units interact pairwise. Many body interactions are often at play, and cannot be modelled by resorting to binary exchanges. In this work, we consider a general class of dynamical systems anchored on hypergraphs. Hyperedges of arbitrary size ideally encircle individual units so as to account for multiple, simultaneous interactions. These latter are mediated by a combinatorial Laplacian, that is here introduced and characterised. The formalism of the Master Stability Function is adapted to the present setting. Turing patterns and the synchronisation of non linear (regular and chaotic) oscillators are studied, for a general class of systems evolving on hypergraphs. The response to externally imposed perturbations bears the imprint of the higher order nature of the interactions.