Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-26T05:17:30.176Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral bounds in random graphs applied to spreading phenomena and percolation

Part of: Graph theory Markov processes Special processes

Published online by Cambridge University Press:  26 July 2018

Rémi Lemonnier ,
Kevin Scaman  and
Nicolas Vayatis
Rémi Lemonnier*
Affiliation:
École Normale Supérieure Paris-Saclay
Kevin Scaman*
Affiliation:
École Normale Supérieure Paris-Saclay
Nicolas Vayatis*
Affiliation:
École Normale Supérieure Paris-Saclay
*
* Postal address: 55 rue de Rome, 75008 Paris, France. Email address: [email protected]
** Postal address: 18 Quai du Point du Jour, 92100 Boulogne-Billancourt, France. Email address: [email protected]
*** Postal address: Centre de Mathématiques et de Leurs Applications (CMLA), École Normale Supérieure Paris-Saclay, 61 Avenue President Wilson, F-94230 Cachan, France. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we derive nonasymptotic upper bounds for the size of reachable sets in random graphs. These bounds are subject to a phase transition phenomenon triggered by the spectral radius of the hazard matrix, a reweighted version of the adjacency matrix. Such bounds are valid for a large class of random graphs, called local positive correlation (LPC) random graphs, displaying local positive correlation. In particular, in our main result we state that the size of reachable sets in the subcritical regime for LPC random graphs is at most of order O(√n), where n is the size of the network, and of order O(n2/3) in the critical regime, where the epidemic thresholds are driven by the size of the spectral radius of the hazard matrix with respect to 1. As a corollary, we also show that such bounds hold for the size of the giant component in inhomogeneous percolation, the SIR model in epidemiology, as well as for the long-term influence of a node in the independent cascade model.

Keywords

Random graphlocal positive correlationhazard matrixpercolationepidemiologyinformation cascade

MSC classification

Primary: 05C80: Random graphs
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 92D30: Epidemiology 60J60: Diffusion processes
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 2 , June 2018 , pp. 480 - 503
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Ben-Naim, E. and Krapivsky, P. L. (2004). Size of outbreaks near the epidemic threshold. Phys. Rev. E 69, 050901. Google Scholar
[2]Bhamidi, S., van der Hofstad, R. and van Leeuwaarden, J. S. H. (2010). Scaling limits for critical inhomogeneous random graphs with finite third moments. Electron. J. Prob. 15, 1682–1702. Google Scholar
[3]Bollobás, B., Janson, S. and Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31, 3–122. Google Scholar
[4]Bollobás, B., Borgs, C., Chayes, J. and Riordan, O. (2010). Percolation on dense graph sequences. Ann. Prob. 38, 150–183. Google Scholar
[5]Chen, W., Wang, Y. and Yang, S. (2009). Efficient influence maximization in social networks. In Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, ACM, New York, pp. 199–208. Google Scholar
[6]Draief, M., Ganesh, A. and Massoulié, L. (2008). Thresholds for virus spread on networks. Ann. Appl. Prob. 18, 359–378. Google Scholar
[7]Erdős, P. and Rényi, A. (1961). On the evolution of random graphs. Bull. Inst. Internat. Statist. 38, 343–347. Google Scholar
[8]Georgii, H.-O., Häggström, O. and Maes, C. (2001). The random geometry of equilibrium phases. In Phase Transitions Critical Phenomena, Vol. 18, Academic Press, San Diego, CA, pp. 1–142. Google Scholar
[9]Gomez-Rodriguez, M. and Schölkopf, B. (2012). Influence maximization in continuous time diffusion networks. In Proceedings of the 28th International Conference on Machine Learning, pp. 313–320. Google Scholar
[10]Gomez-Rodriguez, M., Balduzzi, D. and Schölkopf, B. (2011). Uncovering the temporal dynamics of diffusion networks. In Proceedings of the 28th International Conference on Machine Learning, pp. 561–568. Google Scholar
[11]Holley, R. (1974). Remarks on the FKG inequalities. Commun. Math. Phys. 36, 227–231. Google Scholar
[12]Kempe, D., Kleinberg, J. and Tardos, E. (2003). Maximizing the spread of influence through a social network. In Proceedings of the 9th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, ACM, New York, pp. 137–146. Google Scholar
[13]Kermack, W. O. and McKendrick, A. G. (1932). Contributions to the mathematical theory of epidemics. II. The problem of endemicity. Proc. R. Soc. London A 138, 55–83. Google Scholar
[14]Lemonnier, R., Scaman, K. and Vayatis, N. (2014). Tight bounds for influence in diffusion networks and application to bond percolation and epidemiology. In Advances in Neural Information Processing Systems 27 (NIPS 2014), pp. 846–854. Google Scholar
[15]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
[16]Molloy, M. and Reed, B. (1998). The size of the giant component of a random graph with a given degree sequence. Combin. Prob. Comput. 7, 295–305. Google Scholar
[17]Nelson, K. E. (2007). Epidemiology of infectious disease: general principles. In Infectious Disease Epidemiology Theory and Practice, 2nd edn, Jones and Bartlett, Sudbury, MA, pp. 25–62. Google Scholar
[18]Newman, M. E. J. (2010). Networks: An Introduction. Oxford University Press. Google Scholar
[19]Norros, I. and Reittu, H. (2006). On a conditionally Poissonian graph process. Adv. Appl. Prob. 38, 59–75. Google Scholar
[20]Prakash, B. A.et al. (2012). Threshold conditions for arbitrary cascade models on arbitrary networks. Knowledge Information Syst. 33, 549–575. Google Scholar
[21]Scaman, K., Lemonnier, R. and Vayatis, N. (2015). Anytime influence bounds and the explosive behavior of continuous-time diffusion networks. In Adv. Neural Information Processing Systems 28 (NIPS 2015), pp. 2017–2025. Google Scholar
[22]Van Mieghem, P., Omic, J. and Kooij, R. (2009). Virus spread in networks. IEEE/ACM Trans. Networking 17, 1–14. Google Scholar