Exact Analysis of Synchronizability for Complex Networks using Regular Graphs

TL;DR

This paper derives exact analytical expressions for the synchronizability of complex networks modeled by regular graphs, specifically r-nearest neighbor cycles and tori, aiding in network control and management.

Contribution

It provides novel analytical formulas for the Laplacian eigenvalues and synchronizability of r-nearest neighbor networks, including multi-dimensional tori, which were previously unavailable.

Findings

Analytical expressions match simulation results.

Network dimension and size significantly affect synchronizability.

Generalized formulas for multi-dimensional networks are established.

Abstract

Network synchronization is an emerging phenomenon in complex networks. The spectrum of Laplacian matrix will be immensely helpful for getting the network dynamics information. Especially, network synchronizability is characterized by the ratio of second smallest eigen value to largest eigen value of the Laplacian matrix. We study the synchronization of complex networks modeled by regular graphs. We obtained the analytical expressions for network synchronizability for r-nearest neighbor cycle and r-nearest neighbor torus. We have also derived the generalized expression for synchronizability for m-dimensional r-nearest neigh- bor torus. The obtained analytical results agree with the simulation results and shown the effect of network dimension, number of nodes and overhead on syn- chronizability and connectivity in complex networks. This work provides the basic analytical tools for managing and controlling the synchronization in the fi- nite sized complex networks and also given the generalized expressions for eigen values of Laplacian matrix for multi dimensional r- nearest neighbor networks.