Analysis and control of network synchronizability

TL;DR

This paper explores how network structure influences synchronizability, revealing that adding edges can both increase or decrease it, and proposes a design method for improving synchronizability through inner linking matrices.

Contribution

It identifies key factors affecting network synchronizability and introduces a design method for inner linking matrices to enhance synchronizability, especially for networks with unbounded regions.

Findings

Adding edges can both increase or decrease synchronizability.

Networks with disconnected complementary graphs benefit from added edges.

A design method for inner linking matrices of rank 1 improves synchronizability.

Abstract

In this paper, the investigation is first motivated by showing two examples of simple regular symmetrical graphs, which have the same structural parameters, such as average distance, degree distribution and node betweenness centrality, but have very different synchronizabilities. This demonstrates the complexity of the network synchronizability problem. For a given network with identical node dynamics, it is further shown that two key factors influencing the network synchronizability are the network inner linking matrix and the eigenvalues of the network topological matrix. Several examples are then provided to show that adding new edges to a network can either increase or decrease the network synchronizability. In searching for conditions under which the network synchronizability may be increased by adding edges, it is found that for networks with disconnected complementary graphs, adding edges never decreases their synchronizability. This implies that better understanding and careful manipulation of the complementary graphs are important and useful for enhancing the network synchronizability. Moreover, it is found that an unbounded synchronized region is always easier to analyze than a bounded synchronized region. Therefore, to effectively enhance the network synchronizability, a design method is finally presented for the inner linking matrix of rank 1 such that the resultant network has an unbounded synchronized region, for the case where the synchronous state is an equilibrium point of the network.