Synchronization and Consensus in State-Dependent Networks

We study synchronization and consensus phenomena in state-dependent graphs in which the edges are weighted according to the Hebbian learning rule or its modified version. By exploring the master stability function of the synchronous state, we show that the modified Hebbian function as coupling strength enlarges the stability region of the synchronous state. In terms of consensus, given that the state-dependent weights are always positive, we prove that consensus in a network of multi-agent systems is always reachable. Furthermore, we show that in state-dependent graphs the second smallest eigenvalue of the graph Laplacian matrix has larger values due to the state-dependency, resulting in speed up of the convergence process.