Notes on Resonant and Synchronized States in Complex Networks

Abstract

Synchronization and resonance on networks are among the most striking collective dynamical phenomena. This paper proposes a simple and compact framework for the description of these processes in networks of coupled oscillators with arbitrary topology and local interactions. New evidences of the multiple relationships between coupling types and intensities and the topological structure of the network are provided, with particular reference to synchronization times and speeds. Some new results about the global resonance frequencies of the network as functions of the eigenvalues of the Laplacian matrix are proved and a comprehensive description of how a single node in the network can act as an influencer or as an influenced node, according to its topological location, are further investigated. A straightforward application to a social network confirms the effectiveness of these findings and provides interesting insights into how these phenomena can arise in networks of this nature.