We live in a world of complex networks, e.g. the Internet, WWW, power grids, social networks and etc. Through probing, collecting and analyzing the topology data of complex networks, people found that the traditional ER (Erdos-Renyi) model [1] cannot well simulate some practical networks such as Internet and WWW. For WWW, Barabάsi and Albert found that the degree distribution follows the power-law form p(k)= k -r with r > 2. This feature also appears to exist in many other complex networks and such networks are called scale-free networks. It is interesting but somewhat amazing that Barabάsi and Albert proposed a simple mechanism, called BA model [2] , to explain the generation of scale-free networks. Since then, the study of scale-free networks has attracted wide attention from many different research fields. Our group developed Dolphin System [3, 4] to probe the IPv6 Internet and found that the IPv6 AS (Autonomous System) backbone network was also scale-free with r < 2 [5]. The Cooperative Association for Internet Data Analysis (CAIDA) [6] also developed their own IPv6 probing tool called Scamper [7] . An important characteristic of scale-free networks is the heterogeneity of the degree distribution, which makes the scale-free network tolerant to random failures but extremely vulnerable to malicious attacks. Error tolerance and attack vulnerability are two common and important properties of complex networks [8] . The relative size of the largest cluster S [8, 9] and the average inversed geodesic L -1 [9] are used to characterize the behavior of the network during attacks. We can also introduce the average network efficiency [10] State Key Laboratory of Software Development Environment Beihang University, 100191, Beijing, P.R.China Email: zhaojichang@nlsde.buaa.edu.cn and kexu@nlsde.buaa.edu.cn Error tolerance and attack vulnerability are two common and important properties of complex networks, which are usually used to evaluate the robustness of a network. Recently, much work has been devoted to determining the network design with optimal robustness. However, little attention has been paid to the problem of how to improve the robustness of existing networks. In this paper, we present a new parameter α, called enforcing parameter, to guide the process of enhancing the robustness of scale-free networks by gradually adding new links. Intuitively, α < 0 means the nodes with lower degrees are selected preferentially while the nodes with higher degrees will be more probably selected when α > 0. It is shown both theoretically and experimentally that when α < 0 the attack survivability of the network can be enforced apparently. Then we propose new strategies to enhance the network robustness. Through extensive experiments and comparisons, we conclude that establishing new links between nodes with low degrees can drastically enforce the attack survivability of scale-free networks while having little impact on the error tolerance.
Zhao et al. REF proposed a new parameter  to guide the process of enhancing the robustness and the experiments showed that the effect of enhancement is better when 0   .
Enhancing the robustness of scale-free networks
J. Phys. A: Math. Theor. 42 (2009) 195003
Computer Science Physics