The Annals of Applied Probability

Thresholds for virus spread on networks

Moez Draief, Ayalvadi Ganesh, and Laurent Massoulié

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We study how the spread of computer viruses, worms and other self-replicating malware is affected by the logical topology of the network over which they propagate. We consider a model in which each host can be in one of 3 possible states—susceptible, infected or removed (cured and no longer susceptible to infection). We characterize how the size of the population that eventually becomes infected depends on the network topology. Specifically, we show that if the ratio of cure to infection rates is larger than the spectral radius of the graph, and the initial infected population is small, then the final infected population is also small in a sense that can be made precise. Conversely, if this ratio is smaller than the spectral radius, then we show in some graph models of practical interest (including power law random graphs) that the average size of the final infected population is large. These results yield insights into what the critical parameters are in determining virus spread in networks.

Article information

Ann. Appl. Probab., Volume 18, Number 2 (2008), 359-378.

First available in Project Euclid: 20 March 2008

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 05C80: Random graphs [See also 60B20]
Secondary: 60J85: Applications of branching processes [See also 92Dxx] 90B15: Network models, stochastic

Reed–Frost epidemic random graphs epidemic threshold spectral radius giant component


Draief, Moez; Ganesh, Ayalvadi; Massoulié, Laurent. Thresholds for virus spread on networks. Ann. Appl. Probab. 18 (2008), no. 2, 359--378. doi:10.1214/07-AAP470.

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  • Alon, N. and Spencer, J. H. (2000). The Probabilistic Method. Wiley, New York.
  • Ball, F. (1983). A threshold theorem for the Reed–Frost chain-binomial epidemic. J. Appl. Probab. 20 153–157.
  • Ball, F., Mollison, D. and Scalia-Tomba, G. (1997). Epidemics with two levels of mixing. Ann. Appl. Probab. 7 46–89.
  • Barabási, A.-L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286 509–512.
  • Barbour, A. D. and Utev, S. (2004). Approximating the Reed–Frost epidemic process. Stoch. Proc. Appl. 113 173–197.
  • Bollobás, B. (2001). Random Graphs. Cambridge Univ. Press.
  • Bollobás, B., Janson, S. and Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31 3–122.
  • Bollobás, B. and Riordan, O. (2004). The diameter of a scale-free random graph. Combinatorica 4 5–34.
  • Berger, N., Borgs, C., Chayes, J. and Saberi, A. (2005). On the spread of viruses on the internet. In Proc. Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms 301–310. SIAM, Philadelphia.
  • Chung, F. and Lu, L. (2002). Connected components in random graphs with given degree sequences. Ann. Combinatorics 6 125–145.
  • Chung, F. and Lu, L. (2003). The average distances in random graphs with given expected degrees. Internet Math. 1 91–114.
  • Chung, F., Lu, L. and Vu, V. (2003). Eigenvalues of random power law graphs. Ann. Combinatorics 7 21–33.
  • Costa, M., Castro, M., Crowcroft, J., Rowstron, A., Zhou, L., Zhang, L. and Barham, P. (2005). Vigilante: End-to-end containment of Internet worms. In Proc. Twentieth ACM Symp. on Operating Systems Principles 133–147. ACM Press, New York.
  • Erdős, P. and Renyi, A. (1960). On the evolution of random graphs. Mat Kutato Int. Közl 5 17–60.
  • Faloutsos, M., Faloutsos, P. and Faloutsos, C. (2003). Power laws and the AS-level Internet topology. IEEE/ACM Trans. Netw. 11 514–524.
  • Ganesh, A., Massoulié, L. and Towley, D. (2005). The effect of network topology on the spread of epidemics. In Proceedings IEEE Infocom.
  • Janson, S., Łuczak, T. and Ruciński, A. (2000). Random Graphs. Wiley, New York.
  • Kermack, W. O. and McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics. Proc. Roy. Soc. Lond. A 115 700–721.
  • Lefevre, C. and Utev, S. (1995). Poisson approximation for the final state of a generalized epidemic process. Ann. Probab. 23 1139–1162.
  • Lovász, L. and Szegedy, B. (2004). Limits of dense graph sequences. Technical Report TR-2004-79, Microsoft Research.
  • Molloy, M. and Reed, B. A. (1995). A critical point for random graphs with a given degree sequence. Random Structures Algorithms 6 161–180.
  • Molloy, M. and Reed, B. A. (1998). The size of the largest component of a random graph on a fixed degree sequence. Combin. Probab. Comput. 7 295–306.
  • Pastor-Satorras, R. and Vespignani, A. (2001). Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86 3200–3203.
  • Pastor-Satorras, R. and Vespignani, A. (2002). Epidemic dynamics in finite scale-free networks. Phys. Rev. E 65.
  • Weaver, N., Paxson, V., Staniford, S. and Cunningham, R. (2003). A taxonomy of computer worms. In ACM Workshop on Rapid Malcode (WORM) 11–18. ACM Press, New York.
  • Williamson, M. M. (2002). Throttling viruses: Restricting propagation to defeat malicious mobile code. In Proc. 18th Annual Computer Security Applications Conference 1–61. IEEE Comput. Soc., Washington.