Robustness analysis of bimodal networks in the whole range of degree correlation

We present an exact analysis of the physical properties of bimodal networks specified by the two peak degree distribution fully incorporating the degree-degree correlation between node connections. The structure of the correlated bimodal network is uniquely determined by the Pearson coefficient of the degree correlation, keeping its degree distribution fixed. The percolation threshold and the giant component fraction of the correlated bimodal network are analytically calculated in the whole range of the Pearson coefficient from $-1$ to 1 against two major types of node removal, which are the random failure and the degree-based targeted attack. The Pearson coefficient for next-nearest-neighbor pairs is also calculated, which always takes a positive value even when the correlation between nearest-neighbor pairs is negative. From the results, it is confirmed that the percolation threshold is a monotonically decreasing function of the Pearson coefficient for the degrees of nearest-neighbor pairs increasing from $-1$ and 1 regardless of the types of node removal. In contrast, the node fraction of the giant component for bimodal networks with positive degree correlation rapidly decreases in the early stage of random failure, while that for bimodal networks with negative degree correlation remains relatively large until the removed node fraction reaches the threshold. In this sense, bimodal networks with negative degree correlation are more robust against random failure than those with positive degree correlation.

Figure 1

Sketch of the allowable region in the $\mu -\nu$ plane (gray region). The red dotted line represents a set of bimodal networks with the same degree distribution $P\left(k\right)$ and the different values of the joint probability $Q(q_1,q_2)$.

Figure 2

Domain of the Pearson's coefficient $r$ as a function of the average degree $\langle k\rangle$. The result is for bimodal networks with $m=3$ and $K=10$.

Figure 3

Plot for the values of the Pearson correlation coefficient $r_2$ for the degrees of next-nearest neighbors as a function of the Pearson coefficient $r$ for the degrees of nearest neighbors. All lines are the results for bimodal networks with degree $m=3$ and degree $K=10$ and the lines from bottom to top correspond to the results for the average degree $\langle k\rangle \approx 6.15$, 5.76, 5.38, 5.00, 4.61, 4.23, and 3.84, respectively.

Figure 4

The percolation threshold $p_{\mathrm{c}}$ for random failure as a function of the Pearson coefficient $r$ for various values of the average degree $\langle k\rangle$.

Figure 5

Pearson's $r$ dependence of the percolation threshold $p_{\mathrm{c}}$ for targeted attack. Legend is the same as Fig. 3.

Figure 6

Plots of the relative size $S\left(p\right)$ of the giant component as a function of the remaining node fraction $p$ (a) for random failure and (b) for targeted attack. The lines are the results obtained by solving Eqs. (24) and (25) numerically. Symbols are the results from numerical simulations under the condition of the number of nodes $N=13000$ for bimodal networks with $m=3, K=10$, and $c=0.5$ (i.e., $\langle k\rangle \approx 4.61$). The broken lines, $S\left(p\right)=p$, representing the maximum robustness are also plotted as a guide to the eye in both plots.

Figure 7

Plot of the robustness measure $R$ as a function of the Pearson coefficient $r$ for random failure (black filled square) and targeted attack (red filled circles). The inset displays the $r$ dependence of the total sum $R_{\mathrm{tot}}$ of the robustness measure for random failure and that for targeted attack.