Applying Percolation Theory

  • Terrence J. Moore
  • Jin-Hee ChoEmail author
Part of the Risk, Systems and Decisions book series (RSD)


Unlike the previous chapter where propagation of failures along the dependency links was studied in a qualitative, human-judgment fashion, this chapter offers an approach to analyzing resilience to failure propagation via a rigorous use of percolation theory. In percolation theory, the basic idea is that a node failure or an edge failure (reverse) percolates throughout a network, and, accordingly, the failure affects the connectivity among nodes. The degree of network resilience can be measured by the size of a largest component (or cluster) after a fraction of nodes or edges are removed in the network. In many cybersecurity applications, the underlying ideas of percolation theory have not been much explored. In this chapter, it is explained how percolation theory can be used to measure network resilience in the process of dealing with different types of network failures. It introduces the measurement of adaptability and recoverability in addition to that of fault tolerance as new contributions to measuring network resilience by applying percolation theory.


Percolation theory Fault tolerance Adaptability Recoverability Network resilience Network failures 


  1. Albert, R., Jeong, H., & Barabási, A. L. (2000). Error and attack tolerance of complex networks. Nature, 406, 378–382.CrossRefGoogle Scholar
  2. Avizienis, A., Laprie, J.-C., Randell, B., & Landwehr, C. (2004). Basic concepts and taxonomy of dependable and secure computing. IEEE Transactions on Dependable and Secure Computing, 1(1), 11–33.CrossRefGoogle Scholar
  3. Bagrow, J. P., Lehmann, S., & Ahn, Y.-Y. (2015). Robustness and modular structure in networks. Network Science, 3(4), 509–525.CrossRefGoogle Scholar
  4. Barabási, A.-L. (2016). Network science. Cambridge University Press, Cambridge, UK.Google Scholar
  5. Blume, L., Easley, D., Kleinberg, J., Kleinberg, R., & Tardos, É. (2011). Which Networks are Least Susceptible to Cascading Failures? In IEEE 52nd Annual Symposium on Foundations of Computer Science, pp. 393–402, Palm Springs.Google Scholar
  6. Broadbent, S., & Hammersley, J. (1957). Percolation processes I. Crystals and mazes. Mathematical Proceedings of the Cambridge Philosophical Society, 53(3), 629–641.MathSciNetCrossRefGoogle Scholar
  7. Budak, C., Agrawal, D., & Abbadi, A. E. (2011). Limiting the spread of misinformation in social networks. ACM International World Wide Web Conference.Google Scholar
  8. Callaway, D. S., Newman, M. E. J., Strogatz, S. H., & Watts, D. J. (2000). Network robustness and fragility: Percolation on random graphs. Physical Review Letters, 85(25), 5468–5471.CrossRefGoogle Scholar
  9. Chau, C.-K., Gibbens, R. J., Hancock, R. E., & Towsley, D. (2011). Robust multipath routing in large wireless networks. Shanghai: Proc. of the IEEE INFOCOM.CrossRefGoogle Scholar
  10. Chen, P.-Y., Cheng, S.-M., & Chen, K.-C. (2012). Smart attacks in smart grid communication networks. IEEE Communications Magazine, 50(8), 24–29.CrossRefGoogle Scholar
  11. Cho, J. H., & Gao, J. (2016). Cyber war game in temporal networks. PLoS One, 11(2), e0148674.CrossRefGoogle Scholar
  12. Cho, J. H., Hurley, P., & Xu, H. (2016). Metrics and measurement of trustworthy systems. Baltimore: IEEE Military Communication Conference (MILCOM).CrossRefGoogle Scholar
  13. Cho, J. H., Xu, S., Hurley, P., Mackay, M., & Benjamin, T. (2017). STRAM: Measuring the trustworthiness of computer-based systems, ACM Computing Surveys (under review).Google Scholar
  14. Chung, F. (2014). A brief survey of PageRank algorithms. IEEE Transactions on Network Science and Engineering, 1(1), 38–42.MathSciNetCrossRefGoogle Scholar
  15. Cohen, R., Erez, K., Ben-Avraham, D., & Havlin, S. (2000). Resilience of the internet to random breakdowns. Physical Review Letters, 85(21), 4626–4628.CrossRefGoogle Scholar
  16. Colbourn, C. (1987). Network resilience. SIAM Journal on Algebraic Discrete Methods, 8(3), 404–409.MathSciNetCrossRefGoogle Scholar
  17. Easley, D., & Kleinberg, J. (2010). Networks, crowds, and markets: Reasoning about a highly connected world, chapter 19: Cascading behavior in networks. Cambridge University Press, Cambridge, UK.Google Scholar
  18. Erdös, P., & Rényi, A. (1960). On the evolution of random graphs. Publications of the Mathematical Institute of the Hungarian Academy of Sciences, 5, 17–61.MathSciNetzbMATHGoogle Scholar
  19. Farr, R. S., Harer, J. L., & Fink, T. M. (2014). Easily repairable networks: Reconnecting nodes after damage. Physical Review Letters, 113(13), 138701.CrossRefGoogle Scholar
  20. Freixas, J., & Pons, M. (2008). The influence of the node criticality relation on some measures of component importance. Operations Research Letters, 36(5), 557–560.MathSciNetCrossRefGoogle Scholar
  21. Girvan, M., & Newman, M. E. J. (2002). Community structure in social and biological networks. Proceedings of the National Academy of Sciences, 99(12), 7821–7826.MathSciNetCrossRefGoogle Scholar
  22. Goel, S., Aggarwal, V., Yener, A., & Calderbank, A. R. (2011). The effect of eavesdroppers on network connectivity: A secrecy graph approach. IEEE Transactions on Information Forensics and Security, 6(3), 712–724.CrossRefGoogle Scholar
  23. Haimes, Y. Y. (2009). On the definition of resilience in systems. Risk Analysis, 29(4), 498–501.MathSciNetCrossRefGoogle Scholar
  24. Huang, Z., Wang, C., Nayak, A., & Stojmenovic, I. (2015). Small cluster in cyber physical systems: Network topology, interdependence and cascading failures. IEEE Transactions on Parallel and Distributed Systems, 26(8), 2340–2351.CrossRefGoogle Scholar
  25. Kong, Z., & Yeh, E. M. (2009). Wireless network resilience to degree-dependent and cascading node failures. In 7th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks, pp. 1–6, Seoul.Google Scholar
  26. Linkov, I., Eisenberg, D. A., Plourde, K., Seager, T. P., Allen, J., & Kott, A. (2013). Resilience metrics for cyber systems. Environment Systems and Decisions, 33(4), 471–476.CrossRefGoogle Scholar
  27. Liu, G., Zhang, J., & Chen, G. (2014). An approach to finding the cost-effective immunization targets for information assurance. Decision Support Systems, 67, 40–52.CrossRefGoogle Scholar
  28. Majdandzic, A., Podobnik, B., Buldrev, S. V., Kenett, D. Y., Havlin, S., & Stanley, H. E. (2014). Spontaneous recovery in dynamical networks. Nature Physics, 10, 34–38.CrossRefGoogle Scholar
  29. McAuley, J., & Leskovec, J. (2012). Learning to discover social circles in ego networks. NIPS, 272, 548–556.Google Scholar
  30. Mizutaka, S., & Yakubo, K. (2013). Overload network failures: an approach from the random-walk model. In 2013 International Conference on Signal-Image Technology & Internet-Based Systems, pp. 630–633, Kyoto.Google Scholar
  31. Moore, C., & Newman, M. (2000). Epidemics and percolation in small-world networks. Physical Review E, 61(5), 5678–5682.CrossRefGoogle Scholar
  32. Najjar, W., & Gaudiot, J.-L. (1990). Network resilience: A measure of network fault tolerance. IEEE Transactions on Computers, 39(2), 174–181.CrossRefGoogle Scholar
  33. Newman, M. E. J. (2010a). Networks: An introduction, chapter 16: Percolation and network resilience (1st ed.). Oxford University Press, Oxford, UK.CrossRefGoogle Scholar
  34. Newman, M. E. J. (2010b). Networks: An introduction, chapter 17: Epidemics on networks (1st ed.). Oxford University Press, Oxford, UK.Google Scholar
  35. Newman, M. E. J. (2010c). Networks: An introduction, chapter 6: Measures and metrics (1st ed.). Oxford University Press, Oxford, UK.Google Scholar
  36. Newman, M., & Watts, D. (1999). Scaling and percolation in the small-world network model. Physical Review E, 60(6), 7332–7342.CrossRefGoogle Scholar
  37. Newman, M., & Ziff, R. (2001). Fast Monte Carlo algorithm for site or bond percolation. Physical Review E, 64(1), 016706.CrossRefGoogle Scholar
  38. Palla, G., Derényi, I., Farkas, I., & Vicsek, T. (2005). Uncovering the overlapping community structure of complex networks in nature and society. Nature, 435, 814–818.CrossRefGoogle Scholar
  39. Premm Raj, H., & Narahari, Y. (2012, August). Influence Limitation in Multi-Campaign Social Networks: A Shapley Value Based Approach. In 8th IEEE International Conference on Autonomous Science and Engineering, pp. 735–740, Seoul, Korea.Google Scholar
  40. Shao, S., Huang, X., Stanley, H. E., & Havlin, S. (2015). Percolation of localized attack on complex networks. New Journal of Physics, 17(2), 023049.MathSciNetCrossRefGoogle Scholar
  41. Shekhtman, L., Danziger, M. M., & Havlin, S. (2016). Recent advances on failure and recovery in networks. Chaos, Solitons, and Fractals, 90, 28–36.CrossRefGoogle Scholar
  42. Sterbenz, J. P. G., Hutchison, D., Çetinkaya, E. K., Jabbar, A., Rohrer, J. P., Schöller, M., & Smith, P. (2010). Resilience and survivability in communication networks: Strategies, principles, and survey of disciplines. Computer Networks, 54(8), 1245–1265.CrossRefGoogle Scholar
  43. Sun, L., & Wang, W. (2013). Understanding blackholes in large-scale cognitive radio networks under generic failures (pp. 728–736). Turin: 2013 Proc. IEEE INFOCOM.Google Scholar
  44. Xing, F., & Wang, W. (2008). On the critical phase transition time of wireless multi-hop networks with random failure. Proc. of ACM MobiCom, San Francisco.Google Scholar
  45. Xu, Y., & Wang, W. (2010). Characterizing the spread of correlated failures in large wireless networks (pp. 1–9). San Diego: 2010 Proc. IEEE INFOCOM.Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  1. 1.U.S. Army Research LaboratoryAdelphiUSA

Personalised recommendations