Synchronization on star graph with noise

TL;DR

This paper studies how noise affects synchronization in the Kuramoto model on star graphs, revealing a transition from abrupt to continuous synchronization depending on noise level through analytical and numerical methods.

Contribution

It introduces a generalized self-consistency equation for star graphs in the Kuramoto model, extending previous work on complete graphs and analyzing noise-induced synchronization transitions.

Findings

Crossover from abrupt to continuous synchronization with increasing noise

Derived a closed-form self-consistency equation for star graphs

Validated the transition through numerical and analytical methods

Abstract

We investigate synchronization in the Kuramoto model with noise on a star graph. By revising the case of a complete graph, we propose a closed form of self-consistency equation for the conventional order parameter and generalize it for a star graph. Using the obtained self-consistency equation, we demonstrate that there is a crossover between the abrupt synchronization at small noise and the continuous phase transition for quite large noise. We probe this crossover numerically and analytically.