Complex network topologies and synchronization

Synchronization in networks with different topologies is studied. We show that for a large class of oscillators there exist two classes of networks; class-A: networks for which the condition of stable synchronous state is sigma gamma(2) > a, and class-B: networks for which this condition reads gamma(N)/gamma(2) < b, where a and b are constants that depend on local dynamics, synchronous state and the coupling matrix, but not on the Laplacian matrix of the graph describing the topology of the network. Here gamma(1) = 0 < gamma(2) <= ...gamma(N) are the eigenvalues of the Laplacian matrix, where N is the order of the graph. Synchronization in networks whose topology is described by classical random graphs and power-law random graphs when N -> infinity is investigated in detail.