In this paper, we investigate the connectivity of the scale-free networks and introduce degree variance which describes the numerical distance between degree and the average degree of vertices of the scale-free networks, and the degree variance is looked as a measure on robustness of the scale free networks subjected to failures. According to degree variance, we investigate the robustness of the scale free networks subject to random failures and intentional attacks, and discuss the vertices' ability to defense the cascading failures. Furthermore, when the average degree is given, a scheme of nonlinear integer program is designed to defense the random and intentional attacks according to degree variance. From the scheme, we can obtain the optimal robustness to cascading failures, and the scheme can be adopted to improve robustness of scale-free networks. Crown Copyright
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