Modeling region-based interconnection for interdependent networks

Various real-world networks interact with and depend on each other. The design of the interconnection between interacting networks is one of the main challenges to achieve a robust interdependent network. Due to cost considerations, network providers are inclined to interconnect nodes that are geographically close. Accordingly, we propose two topologies, the random geographic graph and the relative neighborhood graph, for the design of interconnection in interdependent networks that incorporates the geographic location of nodes. Differing from the one-to-one interconnection studied in the literature, one node in one network can depend on an arbitrary number of nodes in the other network. We derive the average number of interdependent links for the two topologies, which enables their comparison. For the two topologies, we evaluate the impact of the interconnection structure on the robustness of interdependent networks against cascading failures. The two topologies are assessed on the real-world coupled Italian Internet and the electric transmission network. Finally, we propose the derivative of the largest mutually connected component with respect to the fraction of failed nodes as a robustness metric. This robustness metric quantifies the damage of the network introduced by a small fraction of initial failures well before the critical fraction of failures at which the whole network collapses.

Figure 1

Two topologies for $B$: (a) random geometric graph and (b) relative neighborhood graph. Since there is no third node in the intersection region (marked as yellow), nodes $i$ and $j$ are connected. Nodes from $G_1$ are represented with filled circles, whereas nodes from $G_2$ are represented with unfilled circles.

Figure 2

Node coordinate.

Figure 3

The probability $p_{ij}\left(r\right)$ that nodes $i$ and $j$ are connected as a function of the radius $r$ in a random geometric graph with $N={10}^4$ nodes.

Figure 4

An example of (a) a set of $N$ nodes and its (b) relative neighborhood graph.

Figure 5

Number of links for $RNG\left(N\right)$ with $N$ ranging from 50 to 200.

Figure 6

An interdependent network with nodes having no interconnected nodes.

Figure 7

Largest mutually connected component as a function of the fraction of removed nodes in interdependent networks. The coupled graphs are complete graphs and the interconnection matrix is $B=J$.

Figure 8

Largest connected component as a function of the fraction of removed nodes in Erdős-Rényi graphs $G_p\left(N\right)$.

Figure 9

Largest mutually connected component as a function of the fraction of removed nodes in interdependent networks. The coupled graphs are Erdős-Rényi random graphs $G_p\left(N\right)$ with $N=50$ and the average degree $E\left[D\right]=6$. The interdependency is one to one. The results are averaged over ${10}^4$ realizations of interdependent graphs.

Figure 10

Largest mutually connected component as a function of the fraction of removed nodes in interdependent networks. The coupled graphs are Erdős-Rényi graphs $G_p\left(N\right)$ with $N=50$ and the average degrees $E\left[D_1\right]=6$ and $E\left[D_2\right]=8$. The interconnection topology is the random geometric graph with $r=0.2$. The results are averaged over ${10}^3$ realizations of interdependent graphs.

Figure 11

Largest mutually connected component as a function of the fraction of removed nodes in interdependent networks. The coupled graphs are Erdős-Rényi graphs $G_p\left(N\right)$ with $N=50$ and the average degree $E\left[D\right]=6$. The radius $r$ in the random geometric graph is ranging from $0.1$ to $\sqrt{2}$. The simulations are averaged over the results from 1000 interdependent graphs.

Figure 12

Largest mutually connected component as a function of the fraction of removed nodes in interdependent networks. The coupled graphs are Barabási-Albert with $N=500$ and the average degree $E\left[D\right]=6$. The radius $r$ in the random geometric graph is ranging from $0.05$ to $\sqrt{2}$. The results are averaged over ${10}^3$ realizations of interdependent graphs.

Figure 13

Largest mutually connected component as a function of the fraction of removed nodes in interdependent networks. The coupled graphs are Erdős-Rényi graphs $G_p\left(N\right)$ with $N=50$ and the average degree $E\left[D\right]=6$. The radius $r$ in the random geometric graph ranges from $0.1$ to $0.16$. The results are averaged over ${10}^3$ realizations of interdependent graphs.

Figure 14

Largest mutually connected component as a function of the fraction of removed nodes in interdependent networks. The coupled graphs are Barabási-Albert with $N=500$ and the average degree $E\left[D\right]=6$. The radius $r$ in the random geometric graph ranges from $0.01$ to $0.04$. The results are averaged over ${10}^3$ realizations of interdependent graphs.

Figure 15

Largest mutually connected component as a function of the fraction of removed nodes in interdependent networks. The coupled graphs are Erdős-Rényi graphs $G_p\left(N\right)$ with $N=50$ and the average degree $E\left[D\right]=6$. The interconnection topology is the relative neighborhood graph. The results are averaged over ${10}^3$ realizations of interdependent graphs.

Figure 16

Largest mutually connected component as a function of the fraction of removed nodes in interdependent networks. The coupled graphs are Barabási-Albert graphs with $N=500$ and the average degree $E\left[D\right]=6$. The interconnection topology is the relative neighborhood graph. The results are averaged over ${10}^3$ realizations of interdependent graphs.

Figure 17

Coupled Italian electrical transmission network (blue) and the Italian backbone of the Internet (red) with the interconnection topology of the random geometric graph.

Figure 18

Coupled Italian electrical transmission network (blue) and the Italian backbone of the Internet (red) with the interconnection topology of the relative neighborhood graph.

Figure 19

Largest mutually connected component as a function of the fraction of removed nodes in interdependent networks. The coupled graphs are the Italian high-bandwidth backbone of the Internet and the Italian high-voltage electrical transmission network. The interconnection topologies are the random geometric graph and the relative neighborhood graph with the same link density.

Figure 20

Largest mutually connected component as a function of the fraction of removed nodes in interdependent networks. The coupled graphs are Erdős-Rényi graphs $G_p\left(N\right)$ with $N=50$ and the average degrees $E\left[D_1\right]=6$ and $E\left[D_2\right]=8$. The interconnection topology is the random geometric graph with $r=0.02$. The results are averaged over ${10}^4$ realizations of interdependent graphs.