Percolation of attack with tunable limited knowledge

Percolation models shed a light on network integrity and functionality and have numerous applications in network theory. This paper studies a targeted percolation ($\alpha$ model) with incomplete knowledge where the highest degree node in a randomly selected set of $n$ nodes is removed at each step, and the model features a tunable probability that the removed node is instead a random one. A “mirror image” process ($\beta$ model) in which the target is the lowest degree node is also investigated. We analytically calculate the giant component size, the critical occupation probability, and the scaling law for the percolation threshold with respect to the knowledge level $n$ under both models. We also derive self-consistency equations to analyze the $k$-core organization including the size of the $k$ core and its corona in the context of attacks under tunable limited knowledge. These percolation models are characterized by some interesting critical phenomena and reveal profound quantitative structure discrepancies between Erdős-Rényi networks and power-law networks.

Figure 1

(a) The relative size of giant component $P_{\infty }$ for ER networks with $\langle q\rangle =5$ and $N={10}^7$ as a function of occupation probability $p$. The points are numerical simulations for the $\alpha$ model with $n=2$ [gradient red (upper) circles] and $n=N$ [gradient red (upper) diamonds] ranging from $\alpha =1,0.8,0.6,0.4,0.2;\beta$ model with $n=2$ [gradient (lower) blue squares] and $n=N$ [gradient blue (lower) pentagrams] ranging from $\beta =1,0.8,0.6,0.4,0.2$. Magenta (middle) triangles are for the case of $\alpha =\beta =0(n=1)$. (b) The critical threshold $p_c$ as a function of $n$ similar for the $\alpha$ model [gradient red (lower) circles] and the $\beta$ model [gradient blue (upper) squares] ranging from $\alpha =1,0.8,0.6,0.4,0.2$ and $\beta =1,0.8,0.6,0.4,0.2$, respectively. (c) The scaling $S$ as a function of $n$ for the $\alpha$ model with $\alpha =1,0.5$ (gradient red circles) and the $\beta$ model with $\beta =1,0.5$ (gradient blue squares). Solid curves are analytical results.

Figure 2

(a) The relative size of giant component $P_{\infty }$ for PL networks with $\gamma =2.2,q_{min}=1,q_{\mathrm{cut}}=3000$, and $N={10}^7$ as a function of occupation probability $p$. The points are numerical simulations for the $\alpha$ model with $n=2$ [gradient red (upper) circles] and $n=N$ [gradient red (upper) diamonds] ranging from $\alpha =1,0.8,0.6,0.4,0.2$; the $\beta$ model with $n=2$ [gradient blue (lower) squares] and $n=N$ [gradient blue (lower) pentagrams] ranging from $\beta =1,0.8,0.6,0.4,0.2$. Magenta (middle) triangles are for the case of $\alpha =\beta =0(n=1)$. (b) The critical threshold $p_c$ as a function of $n$ similarly for the $\beta$ model (gradient blue squares) ranging from $\beta =1,0.8,0.6,0.4,0.2$. For the $\alpha$ model, $p_c=0$ constantly. (c) The scaling $S$ as a function of $n$ for the $\beta$ model with $\beta =1,0.5$ (gradient blue squares). Solid curves are analytical results.

Figure 3

The relative size of $k$-core $N_{kc}$ with (a) $k=2$ and (b) $k=3$ for ER networks with $\langle q\rangle =5$ and $N={10}^7$ as a function of occupation probability $p$. The points are numerical simulations for the $\alpha$ model with $n=2$ [gradient red (upper) circles] and $n=N$ [gradient red (upper) diamonds] ranging from $\alpha =1,0.8,0.6,0.4,0.2;\beta$ model with $n=2$ [gradient blue (lower) squares] and $n=N$ [gradient blue (lower) pentagrams] ranging from $\beta =1,0.8,0.6,0.4,0.2$. The corresponding relative size of corona $N_k\left(k\right)$ similar with (c) $k=2$ and (d) $k=3$ as a function of $p$. Magenta (middle) triangles are for the case of $\alpha =\beta =0(n=1)$. Solid curves are analytical results.

Figure 4

The relative size of $k$-core $N_{kc}$ with (a) $k=2$ and (b) $k=3$ for PL networks with $\gamma =2.2,q_{min}=1,q_{\mathrm{cut}}=3000$, and $N={10}^7$ as a function of occupation probability $p$. The points are numerical simulations for the $\alpha$ model with $n=2$ [gradient red (upper) circles] ranging from (a) $\alpha =1,0.8,0.6,0.4,0.2$ and (b) $\alpha =1,0.5;\beta$ model with $n=2$ [gradient blue (lower) squares] ranging from (a) $\beta =1,0.8,0.6,0.4,0.2$ and (b) $\beta =1,0.5$. The corresponding relative size of corona $N_k\left(k\right)$ with (c) $k=2$ and (d) $k=3$ as a function of $p$. Magenta (middle) triangles are for the case of $\alpha =\beta =0(n=1)$. Solid curves are analytical results.