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.

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It is undisputed that complex networks are used to describe the interaction between large-scale complex systems. Different edges have different effects on network controllability. When some edges in a network are attacked or destroyed, the network controllability may be affected very little; when some other edges are attacked, network controllability may be affected very greatly, even results in the uncontrollability of the network. Which edges failure will affect the network controllability? To solve this problem, according to the node classification and edge classification, the concept of quasi-critical edge set is proposed, and the judgment theorem of quasi-critical edge set is given in this paper. In order to study the influence of quasi-critical edge set on the network controllability, the failure model of quasi-critical edge set is proposed, and the network controllability is quantified by the ratio of the number of driver nodes to the number of network nodes. In this failure model, the quasi-critical edge set with the minimum number of edges is removed first, thus destroying the network controllability quickly. By analyzing the failure model of quasi-critical edge set, the failure curve of quasi-critical edge set is obtained. It is found that the failure curve is a piecewise linear function and that the maximum (initial) slope of failure curve is related to the average degree of network. In addition, the failure of quasi-critical edge set has the greatest influence on network controllability. A comparison among the failure of quasi-critical edge set, random failure, degree failure, and betweenness failure verifies that the failure of quasi-critical edge set has the greatest damage to the network controllability in both model networks (ER random network, BA scale-free network, random triangle network and random rectangle network) and real networks in 26 different fields. For some of real networks, such as cancer cell networks, terrorist communication networks and other networks that are harmful to human beings, the failure model of quasi-critical edge set can provide a reference attack method.