Hostname: page-component-848d4c4894-2xdlg Total loading time: 0 Render date: 2024-06-25T09:16:11.720Z Has data issue: false hasContentIssue false
  • English
  • Français

Poisson Approximation of the Number of Cliques in Random Intersection Graphs

Part of: Limit theorems Graph theory

Published online by Cambridge University Press:  14 July 2016

Katarzyna Rybarczyk  and
Dudley Stark
Katarzyna Rybarczyk*
Affiliation:
Adam Mickiewicz University
Dudley Stark*
Affiliation:
Queen Mary, University of London
*
∗Postal address: Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Umultowska 87, 61-614 Poznań, Poland.
∗∗Postal address: School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS, UK. Email address: d.s.stark@qmul.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A random intersection graph G(n, m, p) is defined on a set V of n vertices. There is an auxiliary set W consisting of m objects, and each vertex v ∈ V is assigned a random subset of objects Wv ⊆ W such that w ∈ Wv with probability p, independently for all v ∈ V and all w ∈ W. Given two vertices v1, v2 ∈ V, we set v1 ∼ v2 if and only if Wv1 ∩ Wv2 ≠ ∅. We use Stein's method to obtain an upper bound on the total variation distance between the distribution of the number of h-cliques in G(n, m, p) and a related Poisson distribution for any fixed integer h.

Keywords

Stein's methodPoisson approximationrandom intersection graph

MSC classification

Secondary: 60F05: Central limit and other weak theorems 05C80: Random graphs
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 3 , September 2010 , pp. 826 - 840
Copyright
Copyright © Applied Probability Trust 2010 

References

[1] Behrisch, M. (2007). Component evolution in random intersection graphs. Electron. J. Combinatorics 14, R17, 12 pp.CrossRefGoogle Scholar
[2] Blackburn, S. R. and Gerke, S. (2009). Connectivity of the uniform random intersection graph. Discrete Math. 309, 5130–5140.CrossRefGoogle Scholar
[3] Blanchard, Ph., Krueger, A., Krueger, T. and Martin, P. (2005). The epidemics of corruption. Preprint. Available at http://arxiv.org/abs/physics/0505031v1.Google Scholar
[4] Bloznelis, M., Jaworski, J. and Rybarczyk, K. (2009). Component evolution in a secure wireless sensor network. Networks 53, 19–26.CrossRefGoogle Scholar
[5] Britton, T., Deijfen, M., Lagerås, A. N. and Lindholm, M. (2008). Epidemics on random graphs with tunable clustering. J. Appl Prob. 45, 743–756.CrossRefGoogle Scholar
[6] Deijfen, M. and Kets, W. (2009). Random intersection graphs with tunable degree distribution and clustering. Prob. Eng. Inf. Sci. 23, 661–674.CrossRefGoogle Scholar
[7] Di Pietro, R. et al. (2006). Sensor networks that are provably resilient. In Proc. IEEE Internat. Conf. on Security and Privacy in Communication Networks (SECURECOMM 2006), Baltimore.Google Scholar
[8] Fill, J. A., Scheinerman, E. R. and Singer-Cohen, K. B. (2000). Random intersection graphs when m=ω(n): an equivalence theorem relating the evolution of the G(n,m,p) and G(n,p) models. Random Structures Algorithms 16, 156–176.3.0.CO;2-H>CrossRefGoogle Scholar
[9] Godehardt, E. and Jaworski, J. (2002). Two models of random intersection graphs for classification. In Exploratory Data Analysis in Empirical Research, eds Opitz, O. and Schwaiger, M., Springer, Berlin, pp. 67–81.Google Scholar
[10] Godehardt, E., Jaworski, J. and Rybarczyk, K. (2007). Random intersection graphs and classification. In Advances in Data Analysis, eds Decker, R. and Lenz, H.-J., Springer, Berlin, pp. 67–74.CrossRefGoogle Scholar
[11] Godehardt, E., Jaworski, J. and Rybarczyk, K. (2010). Isolated vertices in random intersection graphs. In Advances in Data Analysis, Data Handling and Business Intelligence, eds Fink, A. et al., Springer, Berlin, pp. 135–145.Google Scholar
[12] Janson, S., Łuczak, T. and Ruciński, A. (2000). Random Graphs. Wiley-Interscience, New York.CrossRefGoogle Scholar
[13] Karoński, M., Scheinerman, E. R. and Singer-Cohen, K. B. (1999). On random intersection graphs: the subgraph problem. Combinatorics Prob. Comput. 8, 131–159.CrossRefGoogle Scholar
[14] Marchette, D. J. (2004). Random Graphs for Statistical Pattern Recognition. John Wiley, Hoboken, NJ.CrossRefGoogle Scholar
[15] Newman, M. E. J., Strogatz, S. H. and Watts, D. J. (2001). Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64, 026118.CrossRefGoogle ScholarPubMed
[16] Newman, M. E. J., Watts, D. J. and Strogatz, S. H. (2002). Random graph models of social networks. Proc. Nat. Acad. Sci. USA 99, 2566–2572.CrossRefGoogle ScholarPubMed