Synchronization transition of heterogeneously coupled oscillators on scale-free networks

TL;DR

This paper analyzes how heterogeneity in coupling affects the synchronization transition in scale-free networks, revealing eight distinct behaviors and deriving critical exponents through mean-field theory and simulations.

Contribution

It introduces a modified Kuramoto model with degree-dependent coupling on scale-free networks and classifies eight different synchronization transition behaviors based on network parameters.

Findings

Eight different synchronization transition behaviors identified.

Critical exponents derived for each transition type.

Analytic results confirmed by numerical simulations.

Abstract

We investigate the synchronization transition of the modified Kuramoto model where the oscillators form a scale-free network with degree exponent $\lambda$. An oscillator of degree $k_i$ is coupled to its neighboring oscillators with asymmetric and degree-dependent coupling in the form of $\couplingcoeff k_i^{\eta-1}$. By invoking the mean-field approach, we determine the synchronization transition point $J_c$, which is zero (finite) when $\eta > \lambda-2$ ($\eta < \lambda-2$). We find eight different synchronization transition behaviors depending on the values of $\eta$ and $\lambda$, and derive the critical exponents associated with the order parameter and the finite-size scaling in each case. The synchronization transition is also studied from the perspective of cluster formation of synchronized vertices. The cluster-size distribution and the largest cluster size as a function of the system size are derived for each case using the generating function technique. Our analytic results are confirmed by numerical simulations.