skip to main content
research-article

Push- and pull-based epidemic spreading in networks: Thresholds and deeper insights

Published:01 October 2012Publication History
Skip Abstract Section

Abstract

Understanding the dynamics of computer virus (malware, worm) in cyberspace is an important problem that has attracted a fair amount of attention. Early investigations for this purpose adapted biological epidemic models, and thus inherited the so-called homogeneity assumption that each node is equally connected to others. Later studies relaxed this often unrealistic homogeneity assumption, but still focused on certain power-law networks. Recently, researchers investigated epidemic models in arbitrary networks (i.e., no restrictions on network topology). However, all these models only capture push-based infection, namely that an infectious node always actively attempts to infect its neighboring nodes. Very recently, the concept of pull-based infection was introduced but was not treated rigorously. Along this line of research, the present article investigates push- and pull-based epidemic spreading dynamics in arbitrary networks, using a nonlinear dynamical systems approach. The article advances the state-of-the-art as follows: (1) It presents a more general and powerful sufficient condition (also known as epidemic threshold in the literature) under which the spreading will become stable. (2) It gives both upper and lower bounds on the global mean infection rate, regardless of the stability of the spreading. (3) It offers insights into, among other things, the estimation of the global mean infection rate through localized monitoring of a small constant number of nodes, without knowing the values of the parameters.

References

  1. Anderson, R. and May, R. 1991. Infectious Diseases of Humans. Oxford University Press.Google ScholarGoogle Scholar
  2. Bailey, N. 1975. The Mathematical Theory of Infectious Diseases and Its Applications, 2nd Ed. Griffin, London.Google ScholarGoogle Scholar
  3. Barabasi, A. and Albert, R. 1999. Emergence of scaling in random networks. Science 286, 509--512.Google ScholarGoogle ScholarCross RefCross Ref
  4. Barrat, A., Barthlemy, M., and Vespignani, A. 2008. Dynamical Processes on Complex Networks. Cambridge University Press. Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. Callaway, D. S., Newman, M. E. J., Strogatz, S. H., and Watts, D. J. 2000. Network robustness and fragility: Percolation on random graphs. Phys. Rev. Lett. 85, 25, 5468--5471.Google ScholarGoogle ScholarCross RefCross Ref
  6. Chakrabarti, D., Wang, Y., Wang, C., Leskovec, J., and Faloutsos, C. 2008. Epidemic thresholds in real networks. ACM Trans. Inf. Syst. Secur. 10, 4, 1--26. Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. Chandler, D. 1987. Introduction to Modern Statistical Mechanics. Oxford University Press.Google ScholarGoogle Scholar
  8. Chen, Z. and Ji, C. 2005. A self-learning worm using importance scanning. In Proceedings of the ACM Workshop on Rapid Malcode (WORM'05). 22--29. Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. Demers, A., Greene, D., Hauser, C., Irish, W., Larson, J., Shenker, S., Sturgis, H., Swinehart, D., and Terry, D. 1987. Epidemic algorithms for replicated database maintenance. In Proceedings of the ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC'87). 1--12. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. Faloutsos, M., Faloutsos, P., and Faloutsos, C. 1999. On power-law relationships of the internet topology. In Proceedings of the ACM SIGCOMM Conference on Applications, Technologies, Architectures and Protocols for Computer Communication (SIGCOMM'99). 251--262. Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. Ganesh, A., Massoulie, L., and Towsley, D. 2005. The effect of network topology on the spread of epidemics. In Proceedings of the IEEE International Conference on Computer Communications (InfoCom'05).Google ScholarGoogle Scholar
  12. Hethcote, H. 2000. The mathematics of infectious diseases. SIAM Rev. 42, 4, 599--653. Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. Horn, R. and Johnson, C. 1985. Matrix Analysis. Cambridge University Press. Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. Karp, R., Schindelhauer, C., Shenker, S., and Vöcking, B. 2000. Randomized rumor spreading. In Proceedings of the IEEE Symposium on Foundations of Computer Science (FOCS'00). 565--574. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. Kempe, D. and Kleinberg, J. 2002. Protocols and impossibility results for gossip-based communication mechanisms. In Proceedings of the 43rdAnnual IEEE Symposium on Foundations of Computer Science (FOCS'02). 471--480. Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. Kempe, D., Kleinberg, J., and Demers, A. 2001. Spatial gossip and resource location protocols. In Proceedings of the Symposium on the Theory of Computing (STOC'01). 163--172. Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. Kephart, J. and White, S. 1991. Directed-Graph epidemiological models of computer viruses. In Proceedings of the IEEE Symposium on Security and Privacy. 343--361.Google ScholarGoogle Scholar
  18. Kephart, J. and White, S. 1993. Measuring and modeling computer virus prevalence. In Proceedings of the IEEE Symposium on Security and Privacy. 2--15. Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. Kermack, W. and McKendrick, A. 1927. A contribution to the mathematical theory of epidemics. Proc. Roy. Soc. Lond. A115, 700--721.Google ScholarGoogle ScholarCross RefCross Ref
  20. Li, X., Parker, P., and Xu, S. 2007. Towards quantifying the (in)security of networked systems. In Proceedings of the 21stIEEE International Conference on Advanced Information Networking and Applications (AINA'07). 420--427. Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. McKendrick, A. 1926. Applications of mathematics to medical problems. Proc. Edin. Math. Soc. 14, 98--130.Google ScholarGoogle Scholar
  22. Medina, A., Lakhina, A., Matta, I., and Byers, J. 2001. Brite: An approach to universal topology generation. In Proceedings of the International Symposium on Modeling, Analysis, and Simulation of Computer and Telecommunication Systems (MASCOTS'01). 346--356. Google ScholarGoogle ScholarDigital LibraryDigital Library
  23. Molloy, M. and Reed, B. 1995. A critical point for random graphs with a given degree sequence. Rand. Struct. Algor. 6, 161--179. Google ScholarGoogle ScholarDigital LibraryDigital Library
  24. Molloy, M. and Reed, B. 1998. The size of the giant component of a random graph with a given degree sequence. Comb. Probab. Comput. 7, 295--305. Google ScholarGoogle ScholarDigital LibraryDigital Library
  25. Moreno, Y., Pastor-Satorras, R., and Vespignani, A. 2002. Epidemic outbreaks in complex heterogeneous networks. Euro. Phys. J. B26, 521--529.Google ScholarGoogle Scholar
  26. Newman, M. 2003. The structure and function of complex networks. SIAM Rev. 45, 167.Google ScholarGoogle ScholarDigital LibraryDigital Library
  27. Newman, M. 2007. Component sizes in networks with arbitrary degree distributions. Phys. Rev. E76, 4, 045101.Google ScholarGoogle Scholar
  28. Newman, M. 2010. Networks: An Introduction. Oxford University Press. Google ScholarGoogle ScholarDigital LibraryDigital Library
  29. Newman, M. E. J., Strogatz, S. H., and Watts, D. J. 2001. Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E64, 2, 026118.Google ScholarGoogle Scholar
  30. Pastor-Satorras, R. and Vespignani, A. 2001. Epidemic dynamics and endemic states in complex networks. Phys. Rev. E63, 066117.Google ScholarGoogle Scholar
  31. Pastor-Satorras, R. and Vespignani, A. 2002. Epidemic dynamics in finite size scale-free networks. Phys. Rev. E65, 035108.Google ScholarGoogle Scholar
  32. Provos, N., McNamee, D., Mavrommatis, P., Wang, K., and Modadugu, N. 2007. The ghost in the browser analysis of web-based malware. In Proceedings of the 1stWorkshop on Hot Topics in Understanding Botnets (HotBots'07). Google ScholarGoogle ScholarDigital LibraryDigital Library
  33. Shah, D. 2009. Gossip algorithms. Found. Trends Netw. 3, 1, 1--125. Google ScholarGoogle ScholarDigital LibraryDigital Library
  34. Wang, Y., Chakrabarti, D., Wang, C., and Faloutsos, C. 2003. Epidemic spreading in real networks: An eigenvalue viewpoint. In Proceedings of the 22ndIEEE Symposium on Reliable Distributed Systems (SRDS'03). 25--34.Google ScholarGoogle Scholar
  35. Wilf, H. 1994. Generating Functionology. Academic Press. Google ScholarGoogle ScholarDigital LibraryDigital Library
  36. Willinger, W., Alderson, D., and Doyle, J. 2009. Mathematics and the internet: A source of enormous confusion and great potential. Not. Amer. Math. Soc. 56, 5, 286--299.Google ScholarGoogle Scholar
  37. Yoshida, K. 1971. Functional Analysis and Its Applications. Springer.Google ScholarGoogle Scholar
  38. Zou, C., Gao, L., Gong, W., and Towsley, D. 2003. Monitoring and early warning for internet worms. In Proceedings of the ACM Conference on Computer and Communications Security (CCS'03), V. Atluri, Ed. ACM Press, New York, 190--199. Google ScholarGoogle ScholarDigital LibraryDigital Library
  39. Zou, C., Gong, W., and Towsley, D. 2002. Code red worm propagation modeling and analysis. In Proceedings of the 9thACM Conference on Computer and Communications Security (CCS'02). 138--147. Google ScholarGoogle ScholarDigital LibraryDigital Library

Index Terms

  1. Push- and pull-based epidemic spreading in networks: Thresholds and deeper insights

      Recommendations

      Reviews

      Edgar R. Weippl

      The authors of this paper first analyze recent work on the epidemic spreading of malware in arbitrary networks, noting that all previous studies only considered push-based infection (where infected hosts actively infect neighbors). They extend this research to include pull-based infections, such as drive-by downloads. To this end, they introduce three parameters (?, ?, and ?), denoting the probability that a given node (?) suffers a push-based infection, (?) is infected using a pull-based method, or (?) is cured. The researchers then describe a general sufficient condition (also called an epidemic threshold) under which the spreading becomes stable, and a more succinct version that holds true under more specific assumptions. They follow with a theorem for upper and lower bounds of the infection rate, "regardless of the stability of the spreading." The theorems presented in the previous sections are confirmed using simulations. The paper uses datasets from Epinions, a social network, and "a graph representing Enron's internal email communications." Although "both networks exhibit power-law degree distributions," it should be noted that the theorems apply to arbitrary networks as well. The authors then discuss the influence of a node's degree on its infection rate, formulating an approximating equation in which the degree is a nonlinear factor and confirming it via simulation. Furthermore, they examine whether "the mean field approach is applicable in arbitrary networks." This approach is a statistical method in which random variables are replaced with their expected (mean) values. The authors argue that this procedure is indeed viable under the condition that the probability of a push-based infection is very small. Lastly, the authors show that the global mean infection rate can be accurately approximated by monitoring only a small constant number of nodes with average degrees and certain second-order degrees. The target audience for this paper includes graduate students and researchers with a theoretical background in computer science or mathematics. The formalization makes it hard reading for practitioners; simply put, there are lots of formulas. Online Computing Reviews Service

      Access critical reviews of Computing literature here

      Become a reviewer for Computing Reviews.

      Comments

      Login options

      Check if you have access through your login credentials or your institution to get full access on this article.

      Sign in

      Full Access

      • Published in

        cover image ACM Transactions on Autonomous and Adaptive Systems
        ACM Transactions on Autonomous and Adaptive Systems  Volume 7, Issue 3
        September 2012
        130 pages
        ISSN:1556-4665
        EISSN:1556-4703
        DOI:10.1145/2348832
        Issue’s Table of Contents

        Copyright © 2012 ACM

        Publisher

        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 1 October 2012
        • Accepted: 1 June 2011
        • Revised: 1 April 2011
        • Received: 1 October 2010
        Published in taas Volume 7, Issue 3

        Permissions

        Request permissions about this article.

        Request Permissions

        Check for updates

        Qualifiers

        • research-article
        • Research
        • Refereed

      PDF Format

      View or Download as a PDF file.

      PDF

      eReader

      View online with eReader.

      eReader
      About Cookies On This Site

      We use cookies to ensure that we give you the best experience on our website.

      Learn more

      Got it!