Advances in Applied Probability

How clustering affects epidemics in random networks

Emilie Coupechoux and Marc Lelarge

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

Adv. in Appl. Probab., Volume 46, Number 4 (2014), 985-1008.

First available in Project Euclid: 12 December 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability
Secondary: 05C80: Random graphs [See also 60B20] 91D30: Social networks

Contagion threshold diffusion random graph clustering


Coupechoux, Emilie; Lelarge, Marc. How clustering affects epidemics in random networks. Adv. in Appl. Probab. 46 (2014), no. 4, 985--1008. doi:10.1239/aap/1418396240.

Export citation


  • 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.
  • Amini, H. (2010). Bootstrap percolation and diffusion in random graphs with given vertex degrees. Electron. J. Combin. 17, paper 25.
  • 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.
  • 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.
  • 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.
  • 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.
  • Bollobás, B. (2001). Random Graphs (Camb. Stud. Adv. Math. 73), 2nd edn. Cambridge University Press.
  • Borgs, C., Chayes, J., Ganesh, A. and Saberi, A. (2010). How to distribute antidote to control epidemics. Random Structures Algorithms 37, 204–222.
  • Britton, T., Deijfen, M., Lagerås, A. N. and Lindholm, M. (2008). Epidemics on random graphs with tunable clustering. J. Appl. Prob. 45, 743–756.
  • 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.
  • Coupechoux, E. and Lelarge, M. (2012). How clustering affects epidemics in random networks. Preprint. Available at
  • Coupechoux, E. and Lelarge, M. (2014). Contagions in random networks with overlapping communities. Preprint. Available at
  • Deijfen, M. and Kets, W. (2009). Random intersection graphs with tunable degree distribution and clustering. Prob. Eng. Inf. Sci. 23, 661–674.
  • Gilbert, E. N. (1959). Random graphs. Ann. Math. Statist. 30, 1141–1144.
  • Gleeson, J. P., Melnik, S. and Hackett, A. (2010). How clustering affects the bond percolation threshold in complex networks. Phys. Rev. E 81, 066114.
  • Janson, S. (2009). On percolation in random graphs with given vertex degrees. Electron. J. Prob. 14, 87–118.
  • Janson, S. (2009). The probability that a random multigraph is simple. Combin. Prob. Comput. 18, 205–225.
  • Janson, S., Łuczak, T. and Rucinski, A. (2000). Random Graphs. Wiley-Interscience, New York.
  • Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.
  • Lelarge, M. (2009). Efficient control of epidemics over random networks. In SIGMETRICS '09, ACM, New York, pp. 1–12.
  • Lelarge, M. (2012). Diffusion and cascading behavior in random networks. Games Econom. Behav. 75, 752–775.
  • Miller, J. C. (2009). Percolation and epidemics in random clustered networks. Phys. Rev. E 80, 020901.
  • Molloy, M. and Reed, B. (1995). A critical point for random graphs with a given degree sequence. Random Structures Algorithms 6, 161–179.
  • Morris, S. (2000). Contagion. Rev. Econom. Stud. 67, 57–78.
  • Newman, M. E. J. (2003). Properties of highly clustered networks. Phys. Rev. E 68, 026121.
  • Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Rev. 45, 167–256.
  • Newman, M. E. J. (2009). Random graphs with clustering. Phys. Rev. Lett. 103 058701.
  • Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.
  • Trapman, P. (2007). On analytical approaches to epidemics on networks. Theoret. Pop. Biol. 71, 160–173.
  • Watts, D. J. (2002). A simple model of global cascades on random networks. Proc. Nat. Acad. Sci. USA 99, 5766–5771.