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.

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Notes

  1. 1.

    Notice that the number of edges to be added to each step has been calibrated in order to assure that the resulting graph has a number of edges comparable to the graphs obtained with ER and WS models with the same number of vertices.

  2. 2.

    Notice that these results are in line with Chi and Cai (2004).

  3. 3.

    Data are collected by the Office of Airline Information, Bureau of Transportation Statistics, Research and Innovative Technology Administration.

  4. 4.

    The term “enplaned passengers”, widely used in the aviation industry, refers to passengers boarding a plane at a particular airport. Since the majority of airport revenues are generated, directly or indirectly, by enplaned passengers, this number is the most important air traffic metric. Data consider the total number of revenue passengers boarding an aircraft (including originating, stopover, and transfer passengers) in both scheduled and non-scheduled services.

  5. 5.

    Last, FAA categories concern smaller airports that are divided between non-hub and non-primary if they have, respectively, more or less than 10,000 annual passengers.

  6. 6.

    Only White Plans Airport, Teterboro Airport and Memphis are connected to more than 150 other airports

  7. 7.

    However, it is noteworthy that, in case of removal of Seattle–Tacoma Airport, the network remains connected. In this case, 92% of the routes are also covered by Los Angeles International.

  8. 8.

    Fairbanks International Airport and Juneau International are ranked at the 121st and 142nd places in terms of number of passengers according to FAA classification based on 2018 data. These airports are classified as small and non-hub primary, respectively.

  9. 9.

    For instance, Ted Steven’s Anchorage International Airport is connected to other 121 domestic airports. Only 36 of these airports are also connected to Fairbanks International Airport.

  10. 10.

    In this application, we use the local measure for unweighted and undirected graphs. Extensions to the weighted and directed case are provided in Barrat et al. (2004) and Clemente and Grassi (2018).

  11. 11.

    Interesting empirical applications of this measure in order to assess the relevance of a node in a financial network can be found, for instance, in Grassi (2010) and Croci and Grassi (2014).

  12. 12.

    Notice that it represents the strength of a weighted graph where edge’s weight is the number of passengers of a specific route.

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Acknowledgements

We would like to thank the editor, the guest editor of the Special Issue “Dynamics of socio-economic systems” and the anonymous referees for their careful reviews on a previous version of this paper. We also thank the attendants to DySES (Dynamics of Socio Economic Systems) 2018 and Workshop on the Economic Science with Heterogeneous Interacting Agents 2019 for their very constructive comments.

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Correspondence to G. P. Clemente.

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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)

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Clemente, G.P., Cornaro, A. A novel measure of edge and vertex centrality for assessing robustness in complex networks. Soft Comput 24, 13687–13704 (2020). https://doi.org/10.1007/s00500-019-04470-w

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Keywords

  • Robustness
  • Kirchhoff index
  • Complex networks
  • Air transportation networks
  • Spatial economics