Reconstruction of simplex structures based on phase synchronization dynamics

Abstract

<sec>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 into 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. </sec><sec>In this work, we reconstruct the simplex structures of a network based on synchronization dynamics between network nodes. More specifically, we construct a topological structure of network 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. 8). This representation allows for identifying high-order network structure, showing its advantages over 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 the strong correlation between remote synchronization among indirectly interacting nodes and the network's underlying symmetry. This finding reveals the intricate relationship between network structure and the dynamical process. In summary, we propose a powerful tool for analyzing complex networks, in particular uncovering the interplay between network structure and dynamics. We provide novel insights for further exploring and understanding the high-order interactions and the underlying symmetry of complex networks.</sec>