Synchronization of topological signals in higher-order adaptive multilayer network

Abstract: The study of synchronization in complex systems has recently been revolutionized by incorporating higher-order interactions through simplicial complexes. Building in particular upon the higher-order Kuramoto model, which considers oscillators on nodes, links, and higher-dimensional simplices. This work extends the monolayer framework of the higher-order Kuramoto model to multilayer networks where the layers are adaptively coupled through order parameters of the oscillators placed on the simplices. We propose two multilayer architectures: one that allows interactions between signals of the same dimension across layers and the other that permits cross-dimensional interactions. We observe that a higher coupling strength is required for synchronization transitions of the node signals and the projected uplink and downlink signals during adaptation. For example, incorporating node dynamics into link evolution delays the onset of synchronization. This study opens an avenue for understanding complex dynamical processes within interconnected higher-order structures. Finally, we present a comprehensive theoretical framework, first for a bilayer network where layers are random networks treated under the annealed approximation, and then extend the analysis to the case of fully connected layers. The theoretical predictions align remarkably well with numerical simulations, accurately capturing the dynamics of the original model in a globally coupled scenario.