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The Kuramoto model (KM) of coupled phase oscillators on complete, Paley, andErdos-Renyi (ER) graphs is analyzed in this work. As quasirandom graphs, thecomplete, Paley, and ER graphs share many structural properties. For instance,they exhibit the same asymptotics of the edge distributions, homomorphismdensities, graph spectra, and have constant graph limits. Nonetheless, we showthat the asymptotic behavior of solutions in the KM on these graphs can bequalitatively different. Specifically, we identify twisted states, steady statesolutions of the KM on complete and Paley graphs, which are stable for onefamily of graphs but not for the other. On the other hand, we show that thesolutions of the IVPs for the KM on complete and random graphs remain close onfinite time intervals, provided they start from close initial conditions andthe graphs are sufficiently large. Therefore, the results of this paperelucidate the relation between the network structure and dynamics in couplednonlinear dynamical systems. Furthermore, we present new results onsynchronization and stability of twisted states for the KM on Cayley and randomgraphs.

Stability of twisted states in the Kuramoto model on Cayley and random graphs