Onset of Synchronization in Complex Networks of Noisy Oscillators

TL;DR

This paper analyzes how synchronization begins in complex networks of noisy oscillators by deriving a mean-field model and calculating the critical coupling strength, considering degree distribution and noise effects, validated through simulations.

Contribution

It introduces a mean-field approach for networks with degree heterogeneity and noise, deriving the critical coupling for synchronization onset.

Findings

Critical coupling strength depends on noise intensity and degree distribution

Mean-field approximation accurately predicts synchronization onset in dense networks

Method's applicability limited by degree homogeneity and lack of correlations

Abstract

We study networks of noisy phase oscillators whose nodes are characterized by a random degree counting the number of its connections. Both these degrees and the natural frequencies of the oscillators are distributed according to a given probability density. Replacing the randomly connected network by an all-to-all coupled network with weighted edges, allows us to formulate the dynamics of a single oscillator coupled to the mean field and to derive the corresponding Fokker-Planck equation. From the latter we calculate the critical coupling strength for the onset of synchronization as a function of the noise intensity, the frequency distribution and the first two moments of the degree distribution. Our approach is applied to a dense small-world network model, for which we calculate the degree distribution. Numerical simulations prove the validity of the made replacement. We also test the applicability to more sparsely connected networks and formulate homogeneity and absence of correlations in the degree distribution as limiting factors of our approach.