Control core of undirected complex networks

Although more attention has been paid to controlling complex network, it is still lack an efficient method to analyze the controllability of a large undirected network. Here, we establish a structural reduction method (SRM) to get a control core of the original undirected network by a revised leaf removal process. The control core keeps the same controllability as the original network and maintains the one-to-one correspondence of the nodes, which means the driver nodes and controllability analysis of the original network can be done in the control core. With this method, we investigate the structural property of the control core in two artificial networks. Then, we illustrate the high efficiency of SRM in analyzing the controllability of large undirected networks and find that the control core consists of many isolated nodes and smaller components, which greatly improves the efficiency of identifying and analyzing the driver nodes. Finally, we apply SRM on some empirical networks, and find more than 20% nodes and 70% links can be removed, which also confirms the high efficiency of SRM. These findings deepen our understanding of the controllability of undirected networks, and provide an effective method for researching the controllability of the large undirected complex networks.