Abstract: A network consists of nodes and links. In dynamical systems, the nodes are interpreted as individual
dynamical systems whose interactions are described by the links. One important and most studied collective
dynamics on networks is the synchronization. Fully synchronized states where all cells are in synchrony, are rare
instances. The more common phenomenon is partial synchronization where communities or clusters of cells are
synchronized. A synchrony-breaking bifurcation refers to a local bifurcation, where a fully synchronous equilibrium
loses its stability and bifurcates to states of less synchrony. In this talk, we introduce a lattice degree which can be
used to study synchrony-breaking bifurcations, both of steady states and of oscillating states, and to pedict the
existence of these bifurcating branches, together with their multiplicity, symmetry and synchrony types.