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How Clustering Affects Epidemics in Random Networks

Part of: Graph theory Combinatorial probability Mathematical sociology

Published online by Cambridge University Press:  22 February 2016

Emilie Coupechoux  and
Marc Lelarge
Emilie Coupechoux*
Affiliation:
INRIA-ENS
Marc Lelarge*
Affiliation:
INRIA-ENS
*
∗ Current address: Laboratoire I3S, CS 40121, Université Nice Sophia Antipolis, 06903 Sophia Antipolis Cedex, France. Email address: coupecho@i3s.unice.fr
∗∗ Postal address: INRIA-ENS, 23 avenue d'Italie, CS 81321, 75214 Paris Cedex 13, France.
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Abstract

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Motivated by the analysis of social networks, we study a model of random networks that has both a given degree distribution and a tunable clustering coefficient. We consider two types of growth process on these graphs that model the spread of new ideas, technologies, viruses, or worms: the diffusion model and the symmetric threshold model. For both models, we characterize conditions under which global cascades are possible and compute their size explicitly, as a function of the degree distribution and the clustering coefficient. Our results are applied to regular or power-law graphs with exponential cutoff and shed new light on the impact of clustering.

Keywords

Contagion thresholddiffusionrandom graphclustering

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 05C80: Random graphs 91D30: Social networks

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 4 , December 2014 , pp. 985 - 1008
Copyright
© Applied Probability Trust 

Footnotes

This paper is part of the author's PhD thesis done at INRIA-ENS.

References

Acemoglu, D., Ozdaglar, A. and Yildiz, E. (2011). Diffusion of innovations in social networks. In Proc. 50th IEEE Conf. Decision Control, IEEE, New York, pp. 2329–2334.Google Scholar
Amini, H. (2010). Bootstrap percolation and diffusion in random graphs with given vertex degrees. Electron. J. Combin. 17, paper 25.Google Scholar
Ball, F. and Sirl, D. (2012). An SIR epidemic model on a population with random network and household structure, and several types of individuals. Adv. Appl. Prob. 44, 63–86.CrossRefGoogle Scholar
Ball, F., Sirl, D. and Trapman, P. (2009). Threshold behaviour and final outcome of an epidemic on a random network with household structure. Adv. Appl. Prob. 41, 765–796.Google Scholar
Ball, F., Sirl, D. and Trapman, P. (2010). Analysis of a stochastic SIR epidemic on a random network incorporating household structure. Math. Biosci. 224, 53–73.Google Scholar
Blume, L. E. (1995). The statistical mechanics of best-response strategy revision. Evolutionary game theory in biology and economics. Games Econom. Behav. 11, 111–145.Google Scholar
Bollobás, B. (2001). Random Graphs (Camb. Stud. Adv. Math. 73), 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
Borgs, C., Chayes, J., Ganesh, A. and Saberi, A. (2010). How to distribute antidote to control epidemics. Random Structures Algorithms 37, 204–222.Google Scholar
Britton, T., Deijfen, M., Lagerås, A. N. and Lindholm, M. (2008). Epidemics on random graphs with tunable clustering. J. Appl. Prob. 45, 743–756.Google Scholar
Coupechoux, E. and Lelarge, M. (2011). Impact of clustering on diffusions and contagions in random networks. In Proc. 5th Internat. Conf. Network Games Control Optim., IEEE, New York, pp. 1–7.Google Scholar
Coupechoux, E. and Lelarge, M. (2012). How clustering affects epidemics in random networks. Preprint. Available at http://uk.arxiv.org/abs/1202.4974v1.Google Scholar
Coupechoux, E. and Lelarge, M. (2014). Contagions in random networks with overlapping communities. Preprint. Available at http://uk.arxiv.org/abs/1303.4325v2.Google Scholar
Deijfen, M. and Kets, W. (2009). Random intersection graphs with tunable degree distribution and clustering. Prob. Eng. Inf. Sci. 23, 661–674.Google Scholar
Gilbert, E. N. (1959). Random graphs. Ann. Math. Statist. 30, 1141–1144.Google Scholar
Gleeson, J. P., Melnik, S. and Hackett, A. (2010). How clustering affects the bond percolation threshold in complex networks. Phys. Rev. E 81, 066114.Google Scholar
Janson, S. (2009). On percolation in random graphs with given vertex degrees. Electron. J. Prob. 14, 87–118.Google Scholar
Janson, S. (2009). The probability that a random multigraph is simple. Combin. Prob. Comput. 18, 205–225.Google Scholar
Janson, S., Łuczak, T. and Rucinski, A. (2000). Random Graphs. Wiley-Interscience, New York.Google Scholar
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.Google Scholar
Lelarge, M. (2009). Efficient control of epidemics over random networks. In SIGMETRICS '09, ACM, New York, pp. 1–12.Google Scholar
Lelarge, M. (2012). Diffusion and cascading behavior in random networks. Games Econom. Behav. 75, 752–775.CrossRefGoogle Scholar
Miller, J. C. (2009). Percolation and epidemics in random clustered networks. Phys. Rev. E 80, 020901.Google Scholar
Molloy, M. and Reed, B. (1995). A critical point for random graphs with a given degree sequence. Random Structures Algorithms 6, 161–179.Google Scholar
Morris, S. (2000). Contagion. Rev. Econom. Stud. 67, 57–78.Google Scholar
Newman, M. E. J. (2003). Properties of highly clustered networks. Phys. Rev. E 68, 026121.Google Scholar
Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Rev. 45, 167–256.Google Scholar
Newman, M. E. J. (2009). Random graphs with clustering. Phys. Rev. Lett. 103 058701.Google Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.Google Scholar
Trapman, P. (2007). On analytical approaches to epidemics on networks. Theoret. Pop. Biol. 71, 160–173.Google Scholar
Watts, D. J. (2002). A simple model of global cascades on random networks. Proc. Nat. Acad. Sci. USA 99, 5766–5771.Google Scholar