Transition to synchrony in degree-frequency correlated Sakaguchi-Kuramoto model

TL;DR

This paper analyzes how the transition to synchronization occurs in the Sakaguchi-Kuramoto model on complex networks, revealing how network type and parameters influence the nature and critical point of synchronization.

Contribution

It provides analytical self-consistent equations for the model and explores the transition types on different network structures, supported by extensive numerical validation.

Findings

Transition type depends on degree distribution and phase-frustration.

Critical coupling decreases with phase-frustration in SF networks.

Transition is always second order in ER networks regardless of phase-lag.

Abstract

We investigate transition to synchrony in degree-frequency correlated Sakaguchi-Kuramoto (SK) model on complex networks both analytically and numerically. We analytically derive self-consistent equations for group angular velocity and order parameter for the model in the thermodynamic limit. Using the self-consistent equations we investigate transition to synchronization in SK model on uncorrelated scale-free (SF) and Erd\H{o}s-R\'enyi (ER) networks in detail. Depending on the degree distribution exponent ($\gamma$) of SF networks and phase-frustration parameter, the population undergoes from first order transition (explosive synchronization (ES)) to second order transition and vice versa. In ER networks transition is always second order irrespective of the phase-lag parameter. We observe that the critical coupling strength for the onset of synchronization is decreased by phase-frustration parameter in case of SF network where as in ER network, the phase-frustration delays the onset of synchronization. Extensive numerical simulations using SF and ER networks are performed to validate the analytical results. An analytical expression of critical coupling strength for the onset of synchronization is also derived from the self consistent equations considering the vanishing order parameter limit.