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Understanding how the interplay between higher-order and multilayerstructures of interconnections influences the synchronization behaviors ofdynamical systems is a feasible problem of interest, with possible applicationin essential topics such as neuronal dynamics. Here, we provide a comprehensiveapproach for analyzing the stability of the complete synchronization state insimplicial complexes with numerous interaction layers. We show that thesynchronization state exists as an invariant solution and derive the necessarycondition for a stable synchronization state in presence of general couplingfunctions. It generalizes the well-known master stability function scheme tothe higher-order structures with multiple interaction layers. We verify ourtheoretical results by employing them on networks of paradigmatic R\"{o}ssleroscillators and Sherman neuronal models, and demonstrate that the presence ofgroup interactions considerably improves the synchronization phenomenon in themultilayer framework.

Stability of synchronization in simplicial complexes with multiple interaction layers