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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.
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网络 N L $n_{\rm D}$ 边移除比例$ p $后网络的$n_{\rm D}$ p= 0.2 p= 0.5 p= 0.8 随机 按度 按介数 FQ 随机 按度 按介数 FQ 随机 按度 按介数 FQ Electronic Circuits_S208 122 189 0.24 0.33 0.29 0.43 0.53 0.48 0.57 0.56 0.77 0.74 0.84 0.85 0.95 Electronic Circuits_S402 252 399 0.23 0.32 0.30 0.42 0.53 0.52 0.56 0.57 0.77 0.76 0.84 0.84 0.95 Electronic Circuits_S838 512 819 0.23 0.33 0.30 0.42 0.53 0.48 0.55 0.57 0.77 0.74 0.84 0.84 0.95 Animal_Hens 32 496 0.03 0.03 0.25 0.19 0.44 0.03 0.59 0.34 0.69 0.19 0.81 0.63 0.88 Collaboration_in jazz 198 5484 0.01 0.01 0.01 0.14 0.43 0.03 0.02 0.32 0.72 0.08 0.38 0.60 0.91 Joint senate press releases 92 954 0.01 0.01 0.01 0.22 0.48 0.07 0.01 0.42 0.76 0.27 0.36 0.53 0.92 Questionnaire for high tech managers_Advice 21 190 0.05 0.05 0.14 0.19 0.43 0.10 0.38 0.43 0.67 0.29 0.62 0.67 0.86 Questionnaire for high tech managers_Friendship 21 102 0.10 0.19 0.14 0.33 0.52 0.24 0.43 0.67 0.76 0.57 0.71 0.86 0.90 Questionnaire for high tech managers_Reports 21 20 0.76 0.76 0.81 0.81 0.81 0.76 0.81 0.81 0.90 0.86 0.86 0.86 0.95 corporate law partnership_law firm 71 892 0.01 0.03 0.04 0.20 0.46 0.06 0.20 0.31 0.73 0.25 0.56 0.65 0.90 Children's network of friendship_Third grade 22 177 0.05 0.05 0.05 0.18 0.36 0.05 0.27 0.27 0.64 0.36 0.68 0.64 0.86 Children's network of friendship_Fourth grade 24 101 0.04 0.04 0.08 0.17 0.29 0.42 0.25 0.33 0.58 0.50 0.71 0.58 0.83 Children's network of friendship_Fifth grade 22 103 0.05 0.05 0.09 0.23 0.36 0.18 0.23 0.41 0.68 0.50 0.64 0.73 0.86 Questionnaire for bank_Advice-seeking 11 30 0.27 0.36 0.45 0.36 0.55 0.45 0.55 0.55 0.73 0.73 0.73 0.73 0.91 Questionnaire for bank_Satisfying 11 51 0.18 0.27 0.27 0.36 0.55 0.36 0.36 0.55 0.73 0.64 0.73 0.64 0.82 Questionnaire for bank_Confiding 11 27 0.18 0.27 0.27 0.36 0.55 0.36 0.45 0.55 0.73 0.64 0.82 0.82 0.91 Questionnaire for bank_Close-friends 11 20 0.36 0.45 0.36 0.45 0.55 0.55 0.64 0.54 0.82 0.82 0.82 0.73 0.91 Trade goods in different countries_Foods 24 307 0.04 0.04 0.08 0.21 0.38 0.04 0.17 0.33 0.67 0.17 0.67 0.58 0.88 Trade goods in different countries_Crude materials 24 307 0.04 0.04 0.04 0.21 0.38 0.04 0.21 0.29 0.71 0.17 0.54 0.71 0.88 Trade goods in different countries_Minerals 24 135 0.13 0.13 0.17 0.38 0.58 0.29 0.50 0.58 0.79 0.58 0.83 0.88 0.92 Trade goods in different countries_Diplomacy 24 369 0.04 0.04 0.04 0.21 0.29 0.04 0.12 0.33 0.58 0.12 0.63 0.75 0.83 Questionnaire for grade seven students_Get on 29 361 0.03 0.03 0.03 0.21 0.34 0.03 0.10 0.34 0.65 0.10 0.52 0.59 0.90 Questionnaire for grade seven students_Best friends 29 181 0.03 0.03 0.03 0.17 0.45 0.14 0.21 0.45 0.72 0.48 0.59 0.62 0.93 Questionnaire for grade seven students_Work with 29 198 0.03 0.07 0.10 0.21 0.45 0.21 0.21 0.41 0.72 0.41 0.59 0.62 0.90 Friendships among high school boys_1957 73 243 0.18 0.25 0.19 0.29 0.53 0.34 0.34 0.49 0.77 0.63 0.67 0.78 0.92 Friendships among high school boys_1958 73 263 0.15 0.19 0.15 0.25 0.52 0.29 0.27 0.47 0.77 0.55 0.64 0.78 0.93 
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