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A network of coupled dynamical systems is represented by a graph whosevertices represent individual cells and whose edges represent couplings betweencells. Motivated by the impact of synchronization results of the Kuramotonetworks, we introduce the generalized class of Laplacian networks, governed bymappings whose Jacobian at any point is a symmetric matrix with row entriessumming to zero. By recognizing this matrix with a weighted Laplacian of theassociated graph, we derive the optimal estimates of its positive, null andnegative eigenvalues directly from the graph topology. Furthermore, we providea characterization of the mappings that define Laplacian networks. Lastly, wediscuss stability of equilibria inside synchrony subspaces for two types ofLaplacian network on a ring with some extra couplings.

Synchrony patterns in Laplacian networks