Provable Security of Networks

We propose a security hypothesis that a network is secure, if any deliberate attacks of a small number of nodes will never generate a global failure of the network, and a robustness hypothesis that a network is robust, if a small number of random errors will never generate a global failure of the network. Based on these hypotheses, we propose a definition of security and a definition of robustness of networks against the cascading failure models of deliberate attacks and random errors respectively, and investigate the principles of the security and robustness of networks. We propose a security model such that networks constructed by the model are provably secure against any attacks of small sizes under the cascading failure models, and simultaneously follow a power law, and have the small world property with a navigating algorithm of time complex O(log n). It is shown that for any network G constructed from the security model, G satisfies some remarkable topological properties, including: (i) the small community phenomenon, that is, G is rich in communities of the form X of size poly logarithmic in log n with conductance bounded by O( 1 |X|β ) for some constant β, (ii) small diameter ∗State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, P. O. Box 8718, Beijing, 100190, P. R. China. Email: {angsheng, yicheng, zhangw}@ios.ac.cn. Correspondence: {angsheng, yicheng}@ios.ac.cn. Angsheng Li is partially supported by the Hundred-Talent Program of the Chinese Academy of Sciences. All authors are supported by the Grand Project “Network Algorithms and Digital Information” of the Institute of software, Chinese Academy of Sciences, and NSFC grant No. 61161130530.