Stability of synchronization in simplicial complexes with multiple interaction layers

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 Rössler oscillators and Sherman neuronal models, and demonstrate that the presence of group interactions considerably improves the synchronization phenomenon in the multilayer framework.

The Impact of Higher-Order Interactions on the Synchronization of Hindmarsh–Rose Neuron Maps under Different Coupling Functions

“…Remark 10. Although a few studies have focused on higher-order networks [42][43][44], investigation and comparison of pairwise and non-pairwise interactions in different homogeneous and heterogenous cases are elaborated in the presented study. 7a,b and 9a,b shows that the stability region occupies a more significant area of the parameter plane 𝜎 1 -𝜎 2 when inner-linking functions are applied to links instead of electrical synapses.…”

“…Remark 10. Although a few studies have focused on higher-order networks [42][43][44], investigation and comparison of pairwise and non-pairwise interactions in different homogeneous and heterogenous cases are elaborated in the presented study.…”

“…In this study, electrical and chemical higher-order interactions, as well as pairwise electrical connections, were studied, and the necessary conditions for the neurons to achieve synchrony are given analytically and numerically. In a similar study [43], the synchronization of β cells subjected to the two-node and three-node interactions was investigated. This study considered the higher-order chemical and electrical synapses, while the two-node connections were assumed as a hybrid synapse.…”

Global synchronization on time-varying higher-order structures

“…By considering the simultaneous interactions of many agents, higher-order structures, namely hypergraphs [19] and simplicial complexes [20], offer a more comprehensive understanding of complex systems. These higher-order structures have been proven to produce novel features in various dynamical processes, including consensus [21,22], random walks [23,24], pattern formation [14,25,26], synchronization [14,[27][28][29][30][31][32], social contagion and epidemics [33,34]. Nevertheless, the suggested framework is not sufficiently general to describe systems with many-body interactions that vary with time.…”

Collective dynamics of swarmalators with higher-order interactions

“…In this context, the impact of higher-order interactions on the domain of synchronization has been the subject of thorough investigation in recent years [55][56][57][58][59][60][61][62][63][64] . These studies have unveiled that the incorporation of higher-order interactions among dynamic units has the potential to give rise to a plethora of new collective phenomena.…”