Graphop Mean-Field Limits and Synchronization for the Stochastic Kuramoto Model

TL;DR

This paper develops a mean field theory for stochastic Kuramoto oscillator networks with heterogeneous connectivity, deriving an exact critical threshold for synchronization transition and validating it through numerical simulations.

Contribution

It introduces a general mean field framework using graphop theory for heterogeneous networks and provides an exact formula for the critical synchronization threshold.

Findings

The mean field theory accurately predicts the synchronization threshold for dense networks.

Numerical results align well with theoretical predictions in most cases.

Sparse networks pose challenges for the threshold prediction accuracy.

Abstract

Models of coupled oscillator networks play an important role in describing collective synchronization dynamics in biological and technological systems. The Kuramoto model describes oscillator's phase evolution and explains the transition from incoherent to coherent oscillations under simplifying assumptions including all-to-all coupling with uniform strength. Real world networks, however, often display heterogeneous connectivity and coupling weights that influence the critical threshold for this transition. We formulate a general mean field theory (Vlasov-Focker Planck equation) for stochastic Kuramoto-type phase oscillator models, valid for coupling graphs/networks with heterogeneous connectivity and coupling strengths, using graphop theory in the mean field limit. Considering symmetric odd-valued coupling functions, we mathematically prove an exact formula for the critical threshold for the incoherence-coherence transition. We numerically test the predicted threshold using large finite-size representations of the network model. For a large class of graph models, we find that the numerical tests agree very well with the predicted threshold obtained from mean field theory. However, the prediction is more difficult in practice for graph structures that are sufficiently sparse. Our findings open future research avenues toward a deeper understanding of mean-field theories for heterogeneous systems.