Robust-yet-fragile nature of interdependent networks

Interdependent networks have been shown to be extremely vulnerable based on the percolation model. Parshani et al. [Europhys. Lett. 92, 68002 (2010)] further indicated that the more intersimilar networks are, the more robust they are to random failures. When traffic load is considered, how do the coupling patterns impact cascading failures in interdependent networks? This question has been largely unexplored until now. In this paper, we address this question by investigating the robustness of interdependent Erdös-Rényi random graphs and Barabási-Albert scale-free networks under either random failures or intentional attacks. It is found that interdependent Erdös-Rényi random graphs are robust yet fragile under either random failures or intentional attacks. Interdependent Barabási-Albert scale-free networks, however, are only robust yet fragile under random failures but fragile under intentional attacks. We further analyze the interdependent communication network and power grid and achieve similar results. These results advance our understanding of how interdependency shapes network robustness.

Figure 1

Cascading failures of coupled ER random graphs (a,b) and BA scale-free networks (c,d) under random failures (a,c) or intentional attacks (b,d). The network size $N_A=N_B=5000$ and average degree $\langle k_A\rangle =\langle k_B\rangle =6$. The relative size of the giant component $G$ is shown as a function of the tolerance parameter $\alpha$ with different types of coupling patterns, namely assortative coupling, random coupling, and disassortative coupling. Cascading failures of the single ER random graph under random failures ($ER\_RF$) or intentional attacks ($ER\_IA$) and the single BA scale-free network under random failures ($BA\_RF$) or intentional attacks ($BA\_IA$) are also plotted as benchmarks. The insets are the evolution of the robust index $RI$ with the tolerant parameter $\alpha$ for both assortative and disassortative coupling. Interdependent networks and benchmark networks are represented by solid lines and dashed lines, respectively.

Figure 2

The evolution of reachability of coupled ER random graphs and BA scale-free networks. The network size $N_A=N_B=5000$ and average degree $\langle k_A\rangle =\langle k_B\rangle =6$. The scaled average spreading factor $f_s$ is shown as a function of the tolerance parameter $\alpha$ with different types of coupling patterns, namely assortative coupling, random coupling, and disassortative coupling. Cascading failures of the single ER random graph under random failures ($ER\_RF$) or intentional attacks ($ER\_IA$) and the single BA scale-free network under random failures ($BA\_RF$) or intentional attacks ($BA\_IA$) are also plotted as benchmarks. Interdependent networks and benchmark networks are represented by solid lines and dashed lines, respectively.

Figure 3

The degree distribution of the communication network and power grid in Italy. The communication network, with $N_{\text{CNET}}=39$ nodes and average connectivity $\langle k_{\text{CNET}}\rangle \approx 2.564$. Inset: The power grid, with $N_{\text{GRID}}=310$ and $\langle k_{\text{GRID}}\rangle \approx 2.239$.

Figure 4

Cascading failures of the interdependent communication network and power grid in Italy under random failures (a) and intentional attacks (b). The network size $N_{\text{CNET}}=39, N_{\text{GRID}}=310$, and average degree $\langle k_{\text{CNET}}\rangle \approx 2.564, \langle k_{\text{GRID}}\rangle \approx 2.239$. The relative size of the giant component $G$ is shown as a function of the tolerance parameter $\alpha$ with different types of coupling patterns, namely geographically nearest coupling ($GN\_RF, GN\_IA$), assortative coupling ($AS\_RF, AS\_IA$), and disassortative coupling ($DIS\_RF, DIS\_IA$). Interdependent networks and benchmark networks are represented by solid and dashed lines, respectively.