A novel measure of edge and vertex centrality for assessing robustness in complex networks

Abstract

In this work, we propose a novel robustness measure for networks, which we refer to as Effective Resistance Centrality of a vertex (or an edge), defined as the relative drop of the Kirchhoff index due to deletion of this vertex (edge) from the network. Indeed, we provide a local robustness measure, able to catch which is the effect of either a specific vertex or a specific edge on the network robustness. The validness of this new measure is illustrated on some typical graphs and on a wide variety of well-known model networks. Furthermore, we analyse the topology of the US domestic flight connections. In particular, we investigate the role that airports play in maintaining the structure of the entire network.

Appendices

Appendix A

In appendix, we show how bounds (10), (11) and (13) are derived.

As regard to bound (10), in case $e_{i,j}$ is not a bridge and $G_{e_{i,j}}$ is connected, we have that $K(G_{e_{i,j}})$ satisfies the most right inequality in (3). Hence, we can write:

$$\begin{aligned}&R_{K}(e_{i,j}, G) \le \frac{\frac{\left( n-1\right) \left( n^2-3n+4\right) }{2}}{(n-1)+\frac{2}{d_{1}}\left( \frac{n(n-1)}{2}-m\right) } -1\\&\quad =\frac{\left( n^2-3n+4\right) }{2\left( 1+\frac{n}{d_{1}}-\frac{2m}{d_{1}(n-1)}\right) }-1. \end{aligned}$$

It is worth pointing out that bound (10) is always greater than zero. We have indeed that we should prove:

$$\begin{aligned} { \left( n^2-3n+4\right) \ge 2\left( 1+\frac{n}{d_{1}}-\frac{2m}{d_{1}(n-1)}\right) }. \end{aligned}$$

Since the graph is connected, $m \ge (n-1)$. So, we can write: $2\left( 1+\frac{n}{d_{1}}-\frac{2m}{d_{1}(n-1)}\right) \le \left( 2+\frac{2(n-2)}{d_{1}}\right) \le \left( 2+2(n-2)\right) .$ It is easy to prove that $\left( n^2-3n+4\right) \ge \left( 2+2(n-2)\right)$ since n is a positive integer.

Similarly, in case $e_{i,j}$ is not a bridge and $G_{e_{i,j}}$ is a connected graph with $n>3$ and $d_{n} > \lfloor {\frac{n}{2}\rfloor }$, by (4), we can easily derive bound (11) as follows:

$$\begin{aligned} { R_{K}(e_{i,j}, G) \le \frac{\left( 3n-1\right) }{(n-1)+\frac{2}{d_{1}}\left( \frac{n(n-1)}{2}-m\right) } -1. } \end{aligned}$$

Also bound (11) is always greater than zero. Since the graph is connected, the denominator is lower than $\left( 2n-2\right)$. It is easy to prove that the inequality $\left( 3n-1\right) \ge \left( 2n-2\right)$ is always verified.

Now, we focus on bound (13). According to the most right inequality in (3), we have

$$\begin{aligned}&K(G_{v_{i}}) \le \frac{\left( n-2\right) \left( (n-1)^2-3(n-1)+4\right) }{2}\\&\quad = \frac{\left( n-2\right) (n^2-5n+8)}{2}. \end{aligned}$$

Since

$$\begin{aligned} {R_{K}(v_{i},G)=\frac{K^{N}(G_{v_{i}})}{K^{N}(G)}-1=\frac{K(G_{v_{i}})}{K(G)}\frac{n}{n-2}-1} \end{aligned}$$

and by applying the most left inequality in (3) at the denominator, we have

$$\begin{aligned} { R_{K}(v_{i},G)\le \frac{\frac{\left( n-2\right) \left( n^2-5n+8 \right) }{2}}{(n-1)+\frac{2}{d_{1}}\left( \frac{n(n-1)}{2}-m\right) }\frac{n}{n-2}-1} \end{aligned}$$

and the most right inequality in (13) easily follows.

In a similar way, we have:

$$\begin{aligned} {R_{K}(v_{i},G)\ge \frac{\left[ (n-2)+\frac{2}{d_{1}}\left( \frac{(n-1)(n-2)}{2}-m\right) \right] \frac{n}{n-2}}{\frac{\left( n-1\right) \left( n^2-3n+4\right) }{2}}-1} \end{aligned}$$

and, by simple algebra, we get the left-most inequality in (13).

It is worth pointing out that right-most inequality in (13) is always greater than one. We have indeed that we should prove:

$$\begin{aligned} { n\left( n^2-5n+8 \right) \ge 2(n-1)\left( 1+\frac{n}{d_{1}}-\frac{2m}{d_{1}(n-1)}\right) } \end{aligned}$$

Since the graph is connected, $m \ge (n-1)$ and, for $n \ge 3$, we have also $d_{1}\ge 2$. So, we can write: $2(n-1)\left( 1+\frac{n}{d_{1}}-\frac{2m}{d_{1}(n-1)}\right) \le (n-1)\left( 2+\frac{2(n-2)}{d_{1}}\right) \le (n-1)n.$ It is easy to prove that the inequality $n\left( n^2-5n+8 \right) \ge n(n-1)$ can be rewritten as $\left( n^2-6n+9 \right) = \left( n-3\right) ^{2} \ge 0$ that is always satisfied.

Appendix B

See Tables 5 and 6.

Table 5 Large primary hubs according to FAA classification (based on enplanements in 2017)

Table 6 Medium primary hubs according to FAA classification (based on enplanements in 2017)