Synchronization of phase oscillators on complex hypergraphs

TL;DR

This paper develops a framework to analyze how higher-order interactions in complex hypergraphs influence the synchronization behavior of coupled phase oscillators, revealing phenomena like bistability and explosive transitions.

Contribution

It introduces a combined hypergraph generative model and dimensionality reduction approach to derive reduced equations for synchronization analysis.

Findings

Bistability occurs at strong triangle coupling.

Explosive synchronization transitions are observed.

Conditions for bistability depend on hypergraph properties.

Abstract

We study the effect of structured higher-order interactions on the collective behavior of coupled phase oscillators. By combining a hypergraph generative model with dimensionality reduction techniques, we obtain a reduced system of differential equations for the system's order parameters. We illustrate our framework with the example of a hypergraph with hyperedges of sizes 2 (links) and 3 (triangles). For this case, we obtain a set of 2 coupled nonlinear algebraic equations for the order parameters. For strong values of coupling via triangles, the system exhibits bistability and explosive synchronization transitions. We find conditions that lead to bistability in terms of hypergraph properties and validate our predictions with numerical simulations. Our results provide a general framework to study synchronization of phase oscillators in hypergraphs, and they can be extended to hypergraphs with hyperedges of arbitrary sizes, dynamic-structural correlations, and other features.