# Competing synchronization on random networks

## TL;DR

This paper investigates how competing synchronization dynamics in Kuramoto models behave on Erdős–Rényi networks, revealing that network heterogeneity significantly alters phase transitions and states compared to fully connected networks.

## Contribution

It extends the analysis of competing Kuramoto models to ER networks, showing the impact of network degree heterogeneity on synchronization phases and transitions.

## Key findings

- High mean degree ER networks exhibit phase diagrams similar to complete networks.
- Low mean degree ER networks lack traveling wave states, altering phase diagrams.
- Mean-field approximation aligns with simulations only for high mean degree networks.

## Abstract

The synchronization pattern of a fully connected competing Kuramoto model with a uniform intrinsic frequency distribution $g(\omega)$ was recently considered. This competing Kuramoto model assigns two coupling constants with opposite signs, $K_1 < 0$ and $K_2 > 0$, to the $1-p$ and $p$ fractions of nodes, respectively. This model has a rich phase diagram that includes incoherent, $\pi$, and traveling wave (TW) phases and a hybrid phase transition with abnormal properties that occurs through an intermediate metastable $\pi$ state. Here, we consider the competing Kuramoto model on Erd\H{o}s--R\'enyi (ER) random networks. Numerical simulations and the mean-field solution based on the annealed network approximation reveal that in this case, when the mean degree of the random networks is large, the features of the phase diagram and transition types are consistent overall with those on completely connected networks. However, when the mean degree is small, the mean-field solution is not consistent with the numerical simulation results; specifically, the TW state does not occur, and thus the phase diagram is changed, owing to the strong heterogeneity of the local environment. By contrast, for the original Kuramoto oscillators, the annealed mean-field solution is consistent with the numerical simulation result for ER networks.