Remote Synchronization Reveals Network Symmetries and Functional Modules

Abstract

We study a Kuramoto model in which the oscillators are associated with the nodes of a complex network and the interactions include a phase frustration, thus preventing full synchronization. The system organizes into a regime of remote synchronization where pairs of nodes with the same network symmetry are fully synchronized, despite their distance on the graph. We provide analytical arguments to explain this result, and we show how the frustration parameter affects the distribution of phases. An application to brain networks suggests that anatomical symmetry plays a role in neural synchronization by determining correlated functional modules across distant locations.

Frustration tuning and perfect phase synchronization in the Kuramoto-Sakaguchi model

“…(19,20) and expanding in fluctuations we again extract here the zeroeth order equation from the θ i dynamics which turns out to be identical to Eq. (4) from the static system.…”

“…(19,20), we solve the equations numerically for a random regular graph of N = 50 with degrees k i = 4 at coupling σ = 0.35 and a specific realisation of frequencies drawn from [−1, 1], and initial conditions θ i (0), µ i (0). In Fig.…”

“…We indeed find this to be the case, to a degree even more than an equilibrium analysis reveals, where enhancement occurs in two ways: macroscopic synchronisation is found at lower values of critical coupling than for the normal Kuramoto system of Eq. (1); and this synchronisation occurs at collective frequencies Ω which may be different from (both less and greater than)ω, something already observed for the global Kuramoto-Sakaguchi model [17,20]. Synchronisation in this setting requires specifically tuned frustration parameters and one might wonder how appropriate configurations of such parameters may be reached in real world systems.…”

“…Some variations of the model consider random pinning fields [19] while in others the frustrations are either global or random for every node [20,22], or allocated according to a multi-network structure for interacting populations [23,24]. While such additions introduce additional degrees of disorder to the traditional Kuramoto dynamics, we are interested in whether frustrations may be tuned to enhance synchronisation.…”

Cluster synchronization and isolated desynchronization in complex networks with symmetries

“…These anecdotal studies illustrate the interesting types of CS that can occur, and suggest a broader relationship between the network structure and synchronization patterns. Recent studies have begun to draw a connection between network symmetry and CS, although all have considered simple networks where the symmetries are apparent by inspection [12][13][14] . More in-depth studies have been done involving bifurcation phenomena and synchronization in ring and point symmetry networks 15,16 .…”