Figure 1

Optimization against random failures. Plot of $f_r$ the critical fraction of randomly deleted nodes where the giant connected component vanishes versus $⟨k⟩$ the average number of links per node in the network, under an optimally chosen degree distribution—in the sense that it maximizes $f_r$ for a given $⟨k⟩$. A permissible minimum $k_{\ell }=1$ and maximum $k_m=8$ node degree constraint was also imposed. For $⟨k⟩$ below $8/7$ there is no percolation in the network even with no node failures. Maximum robustness occurs when all the nodes have $k_m$ links, in which case the percolation transition occurs at $f_r=6/7$. For comparison purposes $f_r$ and $⟨k⟩$ for two real networks are plotted: *, Western United States electrical power grid, an exponential network; ○, Internet router network, a power-law network. For the power grid $(k_{\ell },k_m)=(1,19)$ and for the internet $(k_{\ell },k_m)=(1,20)$ [24]. The values of $f_r$, $⟨k⟩$, $k_{\ell }$, and $k_m$ for these real networks were computed from data in Refs. 3, 25.Reuse & Permissions

Figure 2

Network under intentional attack. Functional nodes are white and nodes destroyed under attack are black. A link between white nodes can be traversed both ways. The nonfunctionality of a black node is characterized by having links to it become one-way incoming links only; i.e., such links work in the direction leading into the black node, but not going out of the black node (note: a link between two black nodes cannot be traversed either way).Reuse & Permissions

Figure 3

Optimization against intentional attacks. (a) Analogous plot to Fig. 1, except now the nodes are deleted under attack mode, meaning the nodes are removed sequentially in descending order according to their degree, starting with the most connected node. Again, we take $(k_{\ell },k_m)=(1,8)$. *, Western United States electrical power grid; ○, Internet router network. (b) $k^{*}$, one of the two node degrees present (the other being $k_{\ell }$) in the associated two-peak optimal degree distribution against attack. (c) $p_{k^{*}}$, the fraction of nodes with degree $k^{*}$ in the optimal degree distribution. The discontinuity in the variables is a consequence of the discreteness of $k^{*}$.Reuse & Permissions

Figure 4

Simultaneous optimization against intentional attacks and random failures. We take $(k_{\ell },k_m)=(1,8)$. To each combination of desired minimal network percolation thresholds, $f_a$ under attack and $f_r$ under random failures, corresponds an optimal network, i.e., one that also minimizes $⟨k⟩$. (a) These optimal networks can be divided into different qualitative classes, illustrated using different colors: A—Robustness to these $(f_a,f_r)$ pairs is not attainable due to the $k_m$ constraint. B—$f_r$ is the limiting constraint. There are two node degrees present in these networks, $k_{\ell }$ and $k_m$. C—$f_a$ is the limiting constraint. There are at most two distinct node degrees in these networks, $k_{\ell }$ and $k^{*}$. D—Both $f_a$ and $f_r$ affect the optimal degree distribution. These networks still have just two distinct node degrees, $k_{\ell }$ and $k^{*}$ (i.e., the potential third degree, $k_m$, turns out to have zero frequency). E—As in D, both $f_a$ and $f_r$ affect the optimal degree distribution but there are now three distinct node degrees in the network, $k_{\ell }$, $k^{*}$, and $k_m$. (b) Contour plot of $⟨k⟩$ for the optimal networks. The $⟨k⟩=k_m=8$ contour represents the maximum achievable robustness. For comparison, the $(f_r,f_a)$ robustness thresholds of two real networks were plotted: *, Western United States power grid (exponential network); ○, Internet router (power-law network). For the power grid $⟨k⟩=2.7$ and for the internet $⟨k⟩=2.5$. Note how the points fall below the respective optimal $⟨k⟩$ contours.Reuse & Permissions