  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.
