Model reduction for Kuramoto models with complex topologies

Synchronization of coupled oscillators is a ubiquitous phenomenon, occurring in topics ranging from biology and physics to social networks and technology. A fundamental and long-time goal in the study of synchronization has been to find low-order descriptions of complex oscillator networks and their collective dynamics. However, for the Kuramoto model, the most widely used model of coupled oscillators, this goal has remained surprisingly challenging, in particular for finite-size networks. Here, we propose a model reduction framework that effectively captures synchronization behavior in complex network topologies. This framework generalizes a collective coordinates approach for all-to-all networks [G. A. Gottwald, Chaos 25, 053111 (2015)CHAOEH1054-150010.1063/1.4921295] by incorporating the graph Laplacian matrix in the collective coordinates. We first derive low dimensional evolution equations for both clustered and nonclustered oscillator networks. We then demonstrate in numerical simulations for Erdős-Rényi networks that the collective coordinates capture the synchronization behavior in both finite-size networks as well as in the thermodynamic limit, even in the presence of interacting clusters.