title: Graph partitions and cluster synchronization in networks of oscillators.
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.
authors: ICTEAM, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium.
authors: Michael T
authors: MT
authors: Schaub
authors: Center for International Development, Harvard University, Cambridge, Maasachusetts 02138, USA.
authors: Neave
authors: N
authors: O'Clery
authors: Computation and Neural Systems Program, California Institute of Technology, Pasadena, California 91125, USA.
authors: Yazan N
authors: YN
authors: Billeh
authors: ICTEAM, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium.
authors: Jean-Charles
authors: JC
authors: Delvenne
authors: naXys and Department of Mathematics, University of Namur, B-5000 Namur, Belgium.
authors: Renaud
authors: R
authors: Lambiotte
authors: Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom.
authors: Mauricio
authors: M
authors: Barahona