Synchronization in the random field Kuramoto model on complex networks

TL;DR

This paper investigates how random pinning fields influence synchronization in the Kuramoto model on complex networks, revealing diverse phase transition behaviors depending on network topology and field distribution.

Contribution

It introduces an analytical framework combining Ott-Antonsen and annealed-network methods to analyze phase transitions under random fields in complex networks.

Findings

Homogeneous fields induce tricritical points and change the order of phase transitions.

Scale-free networks with 2<γ≤5 exhibit second-order transitions regardless of field strength.

Strong Gaussian fields do not eliminate second-order transitions, but increase critical coupling.

Abstract

We study the impact of random pinning fields on the emergence of synchrony in the Kuramoto model on complete graphs and uncorrelated random complex networks. We consider random fields with uniformly distributed directions and homogeneous and heterogeneous (Gaussian) field magnitude distribution. In our analysis we apply the Ott-Antonsen method and the annealed-network approximation to find the critical behavior of the order parameter. In the case of homogeneous fields, we find a tricritical point above which a second-order phase transition gives place to a first-order phase transition when the network is either fully connected, or scale-free with the degree exponent $\gamma>5$. Interestingly, for scale-free networks with $2<\gamma \leq 5$, the phase transition is of second-order at any field magnitude, except for degree distributions with $\gamma=3$ when the transition is of infinite order at $K_c=0$ independently on the random fields. Contrarily to the Ising model, even strong Gaussian random fields do not suppress the second-order phase transition in both complete graphs and scale-free networks though the fields increase the critical coupling for $\gamma > 3$. Our simulations support these analytical results.