Conditions for the formation of clusters depending on the conductance and the coefficient of clustering

Abstract

One of the ultimate goals of researches on complex networks is to understand how the structure of complex networks affects the dynamical process taking place on them, such as traffic flow, epidemic spread, cascading behavior, and so on. In previous works [1] and [2], we have studied the synchronizability of a network in terms of the local dynamics, supposing that the topology of the graph is fixed. Now, we are interested in studying the effects of the structure of the network, i.e., the topology of the graph on the network synchronizability. The synchronization interval is given by a formula relating the first non zero and the largest eigenvalue of the Laplacian matrix of the graph with the maximum Lyapunov exponent of the local nodes. Our goal is to understand under what conditions can ensure the formation of clusters depending on the conductance and the coefficient of clustering.