Enhancement of Synchronizability in Networks with Community Structure through Adding Efficient Inter-Community Links

In this paper we propose a framework for enhancing synchronizability of networks with community structure through adding efficient inter-community links. Adding new inter-community links to a network with community structure usually improves its synchronizability. However, this is achieved by increasing communication cost in the network. Thus, the links should be designed in a way that adding them results in maximal improvement. Here, we first consider two disjoint communities, and then, propose an algorithm for choosing nodes from each community to create a certain number of inter-community links between them. We propose an efficient rewiring algorithm, which uses a criterion based on eigenvectors corresponding to the largest and second smallest eigenvalues of the Laplacian matrix. The algorithm uses a modified simulated annealing approach for the optimization process. Extensive numerical simulations show that the proposed algorithm can systematically improve the synchronizability of the network as defined by the eigenratio of the Laplacian. We also show that the proposed rewiring algorithm outperforms heuristic methods such as finding the best possible inter-community links between hub nodes, while having much better computational complexity as well. We study the properties of the end-nodes connected to the final optimized inter-community links. We find that in some cases these nodes have much less centrality values (e.g., degree and betweenness) than the hub nodes of the communities, which means that the optimization algorithm finds non-hub nodes to create inter-community connections in between. The optimized networks are also shown to be good structures for phase synchronizability of non-identical oscillators.