In this paper, the relationship between the network synchronizability and the edge distribution of its associated graph is investigated. First, it is shown that adding one edge to a cycle definitely decreases the network sychronizability. Then, since
sometimes the synchronizability can be enhanced by changing the network structure, the question of whether the networks with more edges are easier to synchronize is addressed. It is shown by examples that the answer is negative. This reveals that generally there are redundant edges in a network, which not only make no contributions to synchronization but actually may reduce the synchronizability. Moreover, an example shows that the node betweenness centrality is not always a good indicator for the network synchronizability. Finally, some more examples are presented to illustrate how the network synchronizability varies following the addition of edges, where all the examples show that the network synchronizability globally increases but locally fluctuates as the number of added edges increases.
In this paper, the problem of pinning control for synchronization of complex dynamical networks is discussed. A cost function of the controlled network is defined by the feedback gain and the coupling strength of the network. An interesting result is
that lower cost is achieved by the control scheme of pinning nodes with smaller degrees. Some rigorous mathematical analysis is presented for achieving lower cost in the synchronization of different star-shaped networks. Numerical simulations on some non-regular complex networks generated by the Barabasi-Albert model and various star-shaped networks are shown for verification and illustration.
