Modules as effective nodes in coarse-grained networks of Kuramoto oscillators

Abstract: Most real-world networks exhibit a significant degree of modularity. Understanding the effects of such topology on dynamical processes is pivotal for advances in social and natural sciences. In this work we consider the dynamics of Kuramoto oscillators on modular networks and propose a simple coarse-graining procedure where modules are replaced by effective single oscillators. The method is inspired by EEG measurements, where very large groups of neurons under each electrode are interpreted as single nodes in a correlation network. We expose the interplay between intra-module and inter-module coupling strengths in keeping the coarse-graining process meaningful and show that its accuracy depends on the degree of intra-module synchronization. We show that, when modules are well synchronized, the phase transition from asynchronous to synchronous motion in networks with 2 and 3 modules is very well described by their respective reduced systems, regardless of the network structure connecting the modules. Application of the method to real networks with small modularity coefficients, on the other hand, reveals that the approximation is not accurate, although it still allows for the computation of the critical coupling and the qualitative behavior of the order parameter if the inter-module coupling is large enough.