The Effect of Link Modifications on Network Synchronization

Abstract

A major issue in studying complex network systems, such as neuroscience and power grids, is understanding the response of network dynamics to link modifications. The notion of network G(G, f, H) refers to di↵usively coupled identical oscillators, where isolated dynamics are chosen to be chaotic. As a consequence of the di↵usive nature, a globally synchronized state emerges as an invariant synchronization subspace, and it will be locally stable above critical coupling strength. Furthermore, the real part of the second minimum eigenvalue of the Laplacian matrix is inverse proportional to the critical coupling strength. Thus, we can use it to determine the synchronizability between two networks. Due to the asymmetry of the Laplacian matrix of a directed graph, adding directed links might cause a decrease in the real part of the second minimum eigenvalue of the Laplacian. If, after adding a link to a graph in a given network, the real part of the second minimum eigenvalue of the Laplacian matrix increases, it is called the enhancement of synchronization. Otherwise, it is called the hindrance of synchronization. In this research, we explore how the stability of synchronization at di↵usively coupled oscillators is a↵ected by link modifications for the networks created using particular motifs, i.e., cycle and star motifs. We consider a weakly connected directed graph consisting of two strongly connected components connected by directed link(s) (called cutset). We study the synchronization transitions in such networks when new directed link(s) between the components, in the opposite direction of the cutset, is added and strongly connects the whole network. We explore which properties of underlying graphs and their connected components may hinder or enhance the synchronization.