Theoretical and Computational Studies to Understand the Behavior of the Kuramoto Model in Different Network Topologies | RISE Research

This study investigates how network topology shapes synchronization in the Kuramoto model, a mathematical framework describing coupled oscillators, by comparing the standard all-to-all coupling to two localized topologies: the ring (closed loop, periodic boundary) and the chain (linear, open boundary). Using Euler-integrated simulations, the author tracks the order parameter r, a measure of synchronization from 0 to 1, over time for both identical and heterogeneous (Gaussian-distributed) oscillator frequencies, across increasing oscillator counts (N = 10 to 1000) and small oscillator counts (N = 2 to 6). With identical frequencies, both topologies converge to phase-locked states: a 'twist state' for the ring and a 'smooth gradient' for the chain. With heterogeneous frequencies, both topologies' order parameters converge toward zero in the thermodynamic limit as boundary effects vanish, but for small oscillator counts they diverge, with the ring consistently sustaining slightly higher synchronization than the chain, most notably at N = 4 where the ring's full rotational symmetry contrasts with the chain's competing boundary and interior dynamics. Finally, comparing critical coupling constants, the standard all-to-all model shows a clear synchronization transition at K approximately 1.6, while the ring topology shows no such transition even up to K = 20, since its nearest-neighbor coupling structure prevents heterogeneous frequencies from achieving global synchronization. The paper concludes that network topology fundamentally alters long-term synchronization dynamics, with applications to real-world locally-coupled oscillator systems such as Josephson junctions, circadian rhythms, and neural or neuromuscular signaling, and suggests future work extend the analysis to second-order Kuramoto models incorporating inertia and damping.