Stability of synchronization in simplicial complexes with multiple interaction layers

TL;DR

This paper extends the master stability function framework to analyze the stability of synchronization in multilayer simplicial complexes with higher-order interactions, providing theoretical insights and empirical validation.

Contribution

It introduces a generalized stability analysis method for synchronization in multilayer simplicial complexes, accounting for higher-order interactions and multiple layers.

Findings

Synchronization stability is enhanced by group interactions.

Theoretical results are validated on R"{o}ssler and neuronal models.

Higher-order structures improve multilayer synchronization.

Abstract

Understanding how the interplay between higher-order and multilayer structures of interconnections influences the synchronization behaviors of dynamical systems is a feasible problem of interest, with possible application in essential topics such as neuronal dynamics. Here, we provide a comprehensive approach for analyzing the stability of the complete synchronization state in simplicial complexes with numerous interaction layers. We show that the synchronization state exists as an invariant solution and derive the necessary condition for a stable synchronization state in presence of general coupling functions. It generalizes the well-known master stability function scheme to the higher-order structures with multiple interaction layers. We verify our theoretical results by employing them on networks of paradigmatic R\"{o}ssler oscillators and Sherman neuronal models, and demonstrate that the presence of group interactions considerably improves the synchronization phenomenon in the multilayer framework.