In a complex network, each edge has different functions on controllability of the whole network. A network may be out of control due to failure or attack of some specific edges. Bridges are a kind of key edges whose removal will disconnect a network and increase connected components. Here, we investigate the effects of removing bridges on controllability of network. Various strategies, including random deletion of edges, deletion based on betweenness centrality, and deletion based on degree of source or target nodes, are used to compare with the effect of removing bridges. It is found that the removing bridges strategy is more efficient on reducing controllability than the other strategies of removing edges for ER networks and scale-free networks. In addition, we also found the controllability robustness under edge attack is related to the average degree of complex networks. Therefore, we propose two optimization strategies based on bridges to improve the controllability robustness of complex networks against attacks. The effectiveness of the proposed strategies is demonstrated by simulation results of some model networks. These results are helpful for people to understand and control spreading processes of epidemic across different paths.
Dynamical analysis and control strategies of an SIVS epidemic model with imperfect vaccination on scale-free networks