# Linear Loss Function for the Network Blocking Game: An Efficient Model for Measuring Network Robustness and Link Criticality

## Abstract

In order to design robust networks, first, one has to be able to measure robustness of network topologies. In [1], a game-theoretic model, the network blocking game, was proposed for this purpose, where a network operator and an attacker interact in a zero-sum game played on a network topology, and the value of the equilibrium payoff in this game is interpreted as a measure of robustness of that topology. The payoff for a given pair of pure strategies is based on a loss-in-value function. Besides measuring the robustness of network topologies, the model can be also used to identify critical edges that are likely to be attacked. Unfortunately, previously proposed loss-in-value functions are either too simplistic or lead to a game whose equilibrium is not known to be computable in polynomial time. In this paper, we propose a new, linear loss-in-value function, which is meaningful and leads to a game whose equilibrium is efficiently computable. Furthermore, we show that the resulting game-theoretic robustness metric is related to the Cheeger constant of the topology graph, which is a well-known metric in graph theory.

## Keywords

game theory adversarial games network robustness computational complexity blocking games Cheeger constant## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Gueye, A., Marbukh, V., Walrand, J.C.: Toward a metric for communication network vulnerability to attacks: A game theoretic approach. In: Proc. of the 3rd International ICST Conference on Game Theory for Networks, GameNets 2012, Vancouver, Canada (May 2012)Google Scholar
- 2.Cunningham, W.: Optimal attack and reinforcement of a network. Journal of the ACM 32(3), 549–561 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
- 3.Bauer, D., Broersma, H., Schmeichel, E.: Toughness in graphs–a survey. Graphs and Combinatorics 22(1), 1–35 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
- 4.Laszka, A., Buttyán, L., Szeszlér, D.: Optimal selection of sink nodes in wireless sensor networks in adversarial environments. In: Proc. of the 12th IEEE International Symposium on a World of Wireless, Mobile and Multimedia, WoWMoM 2011, Lucca, Italy, pp. 1–6 (June 2011)Google Scholar
- 5.Gueye, A., Walrand, J.C., Anantharam, V.: Design of Network Topology in an Adversarial Environment. In: Alpcan, T., Buttyán, L., Baras, J.S. (eds.) GameSec 2010. LNCS, vol. 6442, pp. 1–20. Springer, Heidelberg (2010)CrossRefGoogle Scholar
- 6.Gueye, A., Walrand, J.C., Anantharam, V.: A network topology design game: How to choose communication links in an adversarial environment? In: Proc. of the 2nd International ICST Conference on Game Theory for Networks, GameNets 2011, Shanghai, China (April 2011)Google Scholar
- 7.Laszka, A., Szeszlér, D., Buttyán, L.: Game-theoretic robustness of many-to-one networks. In: Proc. of the 3rd International ICST Conference on Game Theory for Networks, GameNets 2012, Vancouver, Canada (May 2012)Google Scholar
- 8.Schwartz, G.A., Amin, S., Gueye, A., Walrand, J.: Network design game with both reliability and security failures. In: Proc. of the 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp. 675–681. IEEE (September 2011)Google Scholar
- 9.Tardos, E.: A strongly polynomial algorithm to solve combinatorial linear programs. Operations Research 34(2), 250–256 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
- 10.Chung, F.R.K.: Spectral graph theory. Number 92 in CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence (1997)zbMATHGoogle Scholar
- 11.Chung, F.R.K.: Laplacians and the Cheeger inequality for directed graphs. Annals of Combinatorics 9(1), 1–19 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
- 12.Alon, N.: On the edge-expansion of graphs. Combinatorics, Probability and Computing 6(2), 145–152 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
- 13.Alon, N.: Spectral techniques in graph algorithms. In: Proc. of the 3rd Latin American Symposium on Theoretical Informatics, Campinas, Brazil, pp. 206–215 (April 1998)Google Scholar
- 14.Mohar, B.: Isoperimetric numbers of graphs. Journal of Combinatorial Theory, Series B 47(3), 274–291 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
- 15.Mohar, B.: Isoperimetric inequalities, growth, and the spectrum of graphs. Linear Algebra and Its Applications 103, 119–131 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
- 16.Bühler, T., Hein, M.: Spectral clustering based on the graph p-Laplacian. In: Proc. of the 26th Annual International Conference on Machine Learning, ICML 2009, Montreal, Canada, pp. 81–88 (June 2009)Google Scholar