Two-community noisy Kuramoto model with general interaction strengths: Part I

TL;DR

This paper extends the noisy Kuramoto model to two communities with distinct intra- and inter-community interaction strengths, providing a detailed analysis of steady states and phase relations.

Contribution

It introduces a geometric approach to analyze the self-consistency equations, classifies parameter regions by solution count, and characterizes phase relations in the two-community model.

Findings

Identified ten parameter regions with maximum solutions

Proved only angles 0 and π are possible between community phases

Derived boundaries for unsynchronized solutions

Abstract

We generalize the study of the noisy Kuramoto model, considered on a network of two interacting communities, to the case where the interaction strengths within and across communities are taken to be different in general. By developing a geometric interpretation of the self-consistency equations, we are able to separate the parameter space into ten regions in which we identify the maximum number of solutions in the steady state. Furthermore, we prove that in the steady-state only the angles 0 and $\pi$ are possible between the average phases of the two communities and derive the solution boundary for the unsynchronized solution. Lastly, we identify the equivalence class relation in the parameter space corresponding to the symmetrically synchronized solution.