Reconstruction of simplex structures based on phase synchronization dynamics

Abstract

High-order interactions as exemplified by simplex and hyper-edge structures have emerged as a prominent area of interest in complex network research. These high-order interactions introduce much complexity to the interplay between nodes, which often require advanced analytical approaches to fully characterize the underlying network structures. For example, methods based on statistical dependencies have been proposed to identify high-order structures from multi-variate time series. In this work, we reconstruct simplex structures of a network by synchronization dynamics between network nodes. More specifically, we construct the network's topological structure by examining the temporal synchronization of phase time series data derived from the Kuramoto-Sakaguchi (KS) model (Fig. 7). In addition, we show that there is an analytical relationship between the Laplacian matrix of the network and phase variables of the linearized KS model. Our method identifies structural symmetric nodes within a network, which therefore builds a correlation between node synchronization behavior and network's symmetry (Fig. S). This representation allows for identifying high order network structure, showing advantages than statistical methods. In addition, remote synchronization is a complex dynamical process where spatially separated nodes within a network can synchronize their states despite the lack of direct interaction. Furthermore, through numerical simulations, we observe a strong correlation between remote synchronization among indirectly interacting nodes and the network's underlying symmetry. This finding underscores the intricate relationship between network structure and the dynamical processes. In summary, we propose a powerful tool for the analysis of complex networks, in particular uncovering the interplay between network structure and dynamics. We provide novel insights for further exploration and understanding of high-order interactions and the underlying symmetry of complex networks.