Abstract
Synchronization over networks depends strongly on the structure of the coupling between the oscillators. When the coupling presents certain regularities, the dynamics can be coarse-grained into clusters by means of External Equitable Partitions of the network graph and their associated quotient graphs. We exploit this graph-theoretical concept to study the phenomenon of cluster synchronization, in which different groups of nodes converge to distinct behaviors. We derive conditions and properties of networks in which such clustered behavior emerges and show that the ensuing dynamics is the result of the localization of the eigenvectors of the associated graph Laplacians linked to the existence of invariant subspaces. The framework is applied to both linear and non-linear models, first for the standard case of networks with positive edges, before being generalized to the case of signed networks with both positive and negative interactions. We illustrate our results with examples of both signed and unsigned graphs for consensus dynamics and for partial synchronization of oscillator networks under the master stability function as well as Kuramoto oscillators.
Highlights
Synchronization phenomena are prevalent in networked systems in biology, physics, and chemistry, as well as in social and technological networks
We show that cluster synchronization can emerge in networks that can be partitioned into groups according to an external equitable partition (EEP) of the graph
External equitable partitions are of interest because the existence of an EEP in a graph has implications for its spectral properties and, for dynamical processes associated with the graph
Summary
Synchronization phenomena are prevalent in networked systems in biology, physics, and chemistry, as well as in social and technological networks The study of these pervasive processes spans many disciplines leading to a rich literature on this subject.. The use of EEPs emphasizes the presence of an invariant subspace in the coupling structure and leads to a coarse-grained description of the network in terms of a quotient graph. This approach complements the grouptheoretical symmetry viewpoint in Refs. We show how the EEP perspective of cluster synchronization can be generalized to signed networks, i.e., graphs with links of positive and negative weights. We show how these results appear for signed networks under the MSF framework as well as for Kuramoto oscillators
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