Small-world networks of Kuramoto oscillators

• The continuum limit for the Kuramoto model on small-world graphs is derived.
• The existence of qq-twisted states, a class of steady-state solutions, is shown.
• The linear stability analysis of the qq-twisted states is performed.
• The analytical estimate of the synchronization rate is obtained.
• A new mechanism of the formation of interfaces is proposed.

The Kuramoto model of coupled phase oscillators on small-world (SW) graphs is analyzed in this work. When the number of oscillators in the network goes to infinity, the model acquires a family of steady state solutions of degree qq, called qq-twisted states. We show that this class of solutions plays an important role in the formation of spatial patterns in the Kuramoto model on SW graphs. In particular, the analysis of qq-twisted states elucidates the role of long-range random connections in shaping the attractors in this model.We develop two complementary approaches for studying qq-twisted states in the coupled oscillator model on SW graphs: linear stability analysis and numerical continuation. The former approach shows that long-range random connections in the SW graphs promote synchronization and yields the estimate of the synchronization rate as a function of the SW randomization parameter. The continuation shows that the increase of the long-range connections results in patterns consisting of one or several plateaus separated by sharp interfaces.These results elucidate the pattern formation mechanisms in nonlocally coupled dynamical systems on random graphs.