Emergence of synchronization in complex networks of interacting dynamical systems

We study the emergence of coherence in large complex networks of interacting heterogeneous dynamical systems. We show that for a large class of dynamical systems and network topologies there is a critical coupling strength at which the systems undergo a transition from incoherent to coherent behavior. We find that the critical coupling strength at which this transition takes place is k c = ( Z λ ) − 1 , where Z depends only on the uncoupled dynamics of the individual systems on each node, while λ is the largest eigenvalue of the network adjacency matrix. Thus we achieve a separation of the problem into two parts, one depending solely on the node dynamics, and one depending solely on network topology.