Hyperedge Overlap drives Synchronizability of Systems with Higher-Order interactions

TL;DR

This paper introduces a framework using hyperedge overlap matrices to analyze how the microscopic organization of higher-order interactions affects the synchronization stability of dynamical systems.

Contribution

It develops a novel hyperedge overlap matrix framework to quantify inter- and intra-order overlaps and their distinct impacts on synchronization in systems with higher-order interactions.

Findings

Large hyperedge overlaps hinder synchronization stability.

Higher interaction order amplifies the effects of hyperedge overlap.

Intra-order overlap promotes disconnected hyperedge sets, reducing synchronization.

Abstract

The microscopic organization of dynamical systems coupled via higher-order interactions plays a pivotal role in understanding their collective behavior. In this paper, we introduce a framework for systematically investigating the impact of the interaction structure on dynamical processes. Specifically, we develop an hyperedge overlap matrix whose elements characterize the two main aspects of the microscopic organization of higher-order interactions: the inter-order hyperedge overlap (non-diagonal matrix elements) and the intra-order hyperedge overlap (encapsulated in the diagonal elements). This way, the first set of terms quantifies the extent of superposition of nodes among hyperedges of different orders, while the second focuses on the number of nodes in common between hyperedges of the same order. Our findings indicate that large values of both types of hyperedge overlap hinder synchronization stability, and that the larger is the order of interactions involved, the more important is their role. Our findings also indicate that the two types of overlap have qualitatively distinct effects on the dynamics of coupled chaotic oscillators. In particular, large values of intra-order hyperedge overlap hamper synchronization by favoring the presence of disconnected sets of hyperedges, while large values of inter-order hyperedge overlap hinder synchronization by increasing the number of shared nodes between groups converging on different trajectories, without necessarily causing disconnected sets of hyperedges.