Finite-size scaling of synchronized oscillation on complex networks

TL;DR

This paper investigates how the synchronization transition in networks of oscillators depends on network size and structure, deriving scaling laws and exponents for different types of networks.

Contribution

It provides a mean-field theoretical framework to analyze finite-size scaling of synchronization on complex networks, including scale-free networks with varying degree heterogeneity.

Findings

Finite size exponent $ar{ u}$ equals 5/2 for $eta>5$ networks.

Exponents $ar{ u}$ and $eta$ depend on $eta$ for $3<eta<5$ networks.

Analytic exponents agree well with numerical simulations.

Abstract

The onset of synchronization in a system of random frequency oscillators coupled through a random network is investigated. Using a mean-field approximation, we characterize sample-to-sample fluctuations for networks of finite size, and derive the corresponding scaling properties in the critical region. For scale-free networks with the degree distribution $P(k)\sim k^{-\gamma}$ at large $k$, we found that the finite size exponent $\bar{\nu}$ takes on the value 5/2 when $\gamma>5$, the same as in the globally coupled Kuramoto model. For highly heterogeneous networks ($3<\gamma <5$), $\bar{\nu}$ and the order parameter exponent $\beta$ depend on $\gamma$. The analytic expressions for these exponents obtained from the mean field theory are shown to be in excellent agreement with data from extensive numerical simulations.