Networks of coupled oscillators appear in an impressive range of systems in nature and technology where they display collective dynamics, such as synchronization.^{1–3} The Kuramoto model describes the phase evolution of oscillators^{4,5} and explains the transition from incoherent to coherent synchronized oscillations for a critical threshold of the coupling strength under simplifying assumptions, such as all-to-all coupling with uniform strength;^{6,7} however, real world networks often display strong heterogeneity in connectivity and coupling strength, which affect the critical threshold.^{8}

We derive a mean-field theory for stochastic Kuramoto-type models and extend it to a large class of heterogeneous graph/network structures via graphop descriptions valid for the mean-field limit.

We prove a mathematically exact formula for the critical threshold, which we test numerically for large finite-size representations of the network model.

M. A. Gkogkas, B. Jüttner, C. Kuehn, E. A. Martens, Graphop mean-field limits and synchronization for the stochastic Kuramoto model, Chaos 32 (2022) 113120