Robustness of a Network of Networks

Network research has been focused on studying the properties of a single isolated network, which rarely exists. We develop a general analytical framework for studying percolation of $n$ interdependent networks. We illustrate our analytical solutions for three examples: (i) For any tree of $n$ fully dependent Erdős-Rényi (ER) networks, each of average degree $\overline{k}$, we find that the giant component is $P_{\infty }=p[1-\exp (-\overline{k}{P}_{\infty }){]}^n$ where $1-p$ is the initial fraction of removed nodes. This general result coincides for $n=1$ with the known second-order phase transition for a single network. For any $n>1$ cascading failures occur and the percolation becomes an abrupt first-order transition. (ii) For a starlike network of $n$ partially interdependent ER networks, $P_{\infty }$ depends also on the topology—in contrast to case (i). (iii) For a looplike network formed by $n$ partially dependent ER networks, $P_{\infty }$ is independent of $n$.

Figure 1

(a) A treelike NON composed of five fully interdependent networks. (b) Graphical representation of a system of interdependent networks as a starlike NON. Circles represent networks, arrows pointing from network $i$ to network 1 represent partial dependency of network 1 on network $i$ indicating that $q_{i1}>0$ fraction of nodes in network 1 depend on nodes in network $i$. (c) A looplike network of networks, where each network depends partially only on one network.

Figure 2

(a)–(c) Loopless fully coupled ($q=1$) NON. (a) The critical fraction $p_c$ for different $\overline{k}$ and $n$ and (b) the minimum average degree ${\overline{k}}_{\min }$ as a function of the number of networks $n$. The results of (a) and (b) are obtained using Eqs. (6, 7) respectively. (c) Fraction of nodes in the mutual giant component $P_{\infty }$ as a function of $p$ for $\overline{k}=5$ and several values of $n$, obtained from Eq. (9). (d) The giant component for a loop type partially dependent NON [Fig. 1c], $P_{\infty }$ as a function of $p$ for $\overline{k}=6$ and two values of $q$, obtained from Eq. (14).

Figure 3

Dependence of the fraction of nodes at the end of the cascade of failures, $P_{\infty }$, on the fraction of nodes, $p$, survived the initial failure for different number of interacting networks, $n$: (a) for a treelike fully interdependent ($q=1$) SF NON with $\lambda =2.5$ and minimum degree 2. (b) for a starlike partially coupled ($q=0.8$) ER NON with $\overline{k}=5$.