Tweaking synchronization by connectivity modifications

Natural and man-made networks often possess locally treelike substructures. Taking such tree networks as our starting point, we show how the addition of links changes the synchronization properties of the network. We focus on two different methods of link addition. The first method adds single links that create cycles of a well-defined length. Following a topological approach, we introduce cycles of varying length and analyze how this feature, as well as the position in the network, alters the synchronous behavior. We show that in particular short cycles can lead to a maximum change of the Laplacian's eigenvalue spectrum, dictating the synchronization properties of such networks. The second method connects a certain proportion of the initially unconnected nodes. We simulate dynamical systems on these network topologies, with the nodes' local dynamics being either discrete or continuous. Here our main result is that a certain number of additional links, with the relative position in the network being crucial, can be beneficial to ensure stable synchronization.