Synchronization in complex networks with long-range interactions

Variants of collective behavior can materialize in large ensembles of coupled dynamical systems, and synchronization is one of the most significant among them due to its enormous applicability from neuronal networks to finance. At the same time, current study of long-range interactions is attracting researchers' attention mainly because interactions among dynamical units in a network may not be present only in the form of short-range direct communications, but also through the long-range connections arising along the long-distant paths among the nodes. Despite a few recent works on synchronization in long-range interacting systems, there are still a lot of areas regarding the influences of long-range communications on top of non-regular complex networks that remain unexplored. Here we derive local and global asymptotic stability conditions for complete synchronization manifold with k-path Laplacian matrices. Importantly, we show that the analytical findings are in excellent agreement with the numerical results. For the numerical illustrations, we contemplate with the Erdos-Renyi random network by means of a long-range connection governed by the power-law and demonstrate the emergence of complete synchronization. We particularly examine the synergy between the coupling strength and the power-law exponent.