Stability and Topology of Scale-Free Networks under Attack and Defense Strategies

We study tolerance and topology of random scale-free networks under attack and defense strategies that depend on the degree $k$ of the nodes. This situation occurs, for example, when the robustness of a node depends on its degree or in an intentional attack with insufficient knowledge of the network. We determine, for all strategies, the critical fraction $p_c$ of nodes that must be removed for disintegrating the network. We find that, for an intentional attack, little knowledge of the well-connected sites is sufficient to strongly reduce $p_c$. At criticality, the topology of the network depends on the removal strategy, implying that different strategies may lead to different kinds of percolation transitions.

Figure 1

The ratio of nonspanning configurations vs the fraction of removed nodes $p$. Lines are simulation data, from networks of $N={10}^6$ nodes, while the circles are the theoretical critical points. (a) Results from 100 different realizations of networks with $\gamma =2.5$. Left to right: $\alpha =4$, 1, 0.5, 0.25, and 0. (b) Results from 300 realizations of networks with $\gamma =3.5$. Left to right: $\alpha =4$, 1, 0.5, 0, $-0.5$, and $-1$.

Figure 2

(a) Values of $p_c$ vs $\gamma$ for different $\alpha$ values: (bottom to top) $\alpha =4$, 1, 0.5, 0, $-0.5$, and $-1$. The symbols represent simulation data ($N={10}^6$ nodes) from 100–300 different network realizations. The solid lines are the theoretical predictions for finite-size networks, while the dashed lines correspond to infinite-size networks. Lower cutoff: $m=1$. (b) Values of $p_c$ vs $\alpha$ for infinite-size networks and different $\gamma$ values. Lower cutoff $m=1$. (c) The same as (a), with a lower cutoff $m=2$. (d) The same as (b), with a lower cutoff $m=2$.

Figure 3

Average shortest distance $⟨l⟩$ between any two nodes of the giant cluster at criticality as a function of the cluster size $N_c$. The results correspond to networks of initially $N={10}^4$, ${10}^5$, and ${10}^6$ nodes. One thousand different configurations have been used for each $N$, except for $N={10}^6$ (100 configurations). The data have been logarithmically binned, and the results have been vertically shifted for presentation clarity. The values of $\alpha$ and $\gamma$ are shown in the plot. The lines represent the theoretical slopes of (top to bottom) $1/2$, $1/3.5$, $1/3$, and $1/2.33$, respectively. Inset: semilogarithmic plot for the case of $\gamma =2.5$ and $\alpha =0$.