Moment-Based Analysis of Synchronization in Small-World Networks of Oscillators

TL;DR

This paper introduces a novel moment-based analytical approach to predict synchronization in small-world networks of oscillators by analyzing the eigenvalue distribution of the Laplacian matrix.

Contribution

It provides the first analytical expressions for the moments of the Laplacian eigenvalue distribution in small-world networks, linking network structure to synchronization stability.

Findings

Analytical moments accurately predict synchronization thresholds.

Eigenvalue distribution moments depend on shortcut probability and connectivity.

Predictions validated through numerical simulations.

Abstract

In this paper, we investigate synchronization in a small-world network of coupled nonlinear oscillators. This network is constructed by introducing random shortcuts in a nearest-neighbors ring. The local stability of the synchronous state is closely related with the support of the eigenvalue distribution of the Laplacian matrix of the network. We introduce, for the first time, analytical expressions for the first three moments of the eigenvalue distribution of the Laplacian matrix as a function of the probability of shortcuts and the connectivity of the underlying nearest-neighbor coupled ring. We apply these expressions to estimate the spectral support of the Laplacian matrix in order to predict synchronization in small-world networks. We verify the efficiency of our predictions with numerical simulations.