Kuramoto model with frequency-degree correlations on complex networks

TL;DR

This paper investigates how correlations between natural frequencies and degrees in complex networks affect the synchronization transition in the Kuramoto model, revealing first-order, second-order, and hybrid phase transitions depending on network properties.

Contribution

It provides analytical and numerical analysis of the Kuramoto model with frequency-degree correlations, identifying different types of phase transitions in scale-free and star networks.

Findings

First-order phase transition in uncorrelated scale-free networks with 2<γ<3

Second-order phase transition for γ>3

Hybrid transition at γ=3 with critical fluctuations

Abstract

We study the Kuramoto model on complex networks, in which natural frequencies of phase oscillators and the vertex degrees are correlated. Using the annealed network approximation and numerical simulations we explore a special case in which the natural frequencies of the oscillators and the vertex degrees are linearly coupled. We find that in uncorrelated scale-free networks with the degree distribution exponent $2 < \gamma < 3$, the model undergoes a first-order phase transition, while the transition becomes of the second order at $\gamma>3$. If $\gamma=3$, the phase synchronization emerges as a result of a hybrid phase transition that combines an abrupt emergence of synchronization, as in first-order phase transitions, and a critical singularity, as in second-order phase transitions. The critical fluctuations manifest themselves as avalanches in synchronization process. Comparing our analytical calculations with numerical simulations for Erd\H{o}s--R\'{e}nyi and scale-free networks, we demonstrate that the annealed network approach is accurate if the the mean degree and size of the network are sufficiently large. We also study analytically and numerically the Kuramoto model on star graphs and find that if the natural frequency of the central oscillator is sufficiently large in comparison to the average frequency of its neighbors, then synchronization emerges as a result of a first-order phase transition. This shows that oscillators sitting at hubs in a network may generate a discontinuous synchronization transition.