Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-16T04:09:31.115Z Has data issue: false hasContentIssue false
  • English
  • Français

RANDOM INTERSECTION GRAPHS WITH TUNABLE DEGREE DISTRIBUTION AND CLUSTERING

Published online by Cambridge University Press:  14 July 2009

Maria Deijfen  and
Willemien Kets
Maria Deijfen
Affiliation:
Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden E-mail: mia@math.su.se
Willemien Kets
Affiliation:
Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501 andCentER, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, Netherlands E-mail: willemien.kets@santafe.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

A random intersection graph is constructed by assigning independently to each vertex a subset of a given set and drawing an edge between two vertices if and only if their respective subsets intersect. In this article a model is developed in which each vertex is given a random weight and vertices with larger weights are more likely to be assigned large subsets. The distribution of the degree of a given vertex is characterized and is shown to depend on the weight of the vertex. In particular, if the weight distribution is a power law, the degree distribution will be as well. Furthermore, an asymptotic expression for the clustering in the graph is derived. By tuning the parameters of the model, it is possible to generate a graph with arbitrary clustering, expected degree, and—in the power-law case—tail exponent.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 23 , Issue 4 , October 2009 , pp. 661 - 674
Copyright
Copyright © Cambridge University Press 2009

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.Bollobás, B., Janson, S. & Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures & Algorithms 31: 3–122.CrossRefGoogle Scholar
2.Britton, T., Deijfen, M., Lagerås, A. & Lindholm, M. (2008). Epidemics on random graphs with tunable clustering. Journal of Applied Probability 45: 743–756.CrossRefGoogle Scholar
3.Britton, T., Deijfen, M., & Martin-Löf, A. (2006). Generating simple random graphs with prescribed degree distribution. Journal of Statistical Physics 124: 1377–1397.CrossRefGoogle Scholar
4.Chung, F. & Lu, L. (2002). Connected components in random graphs with given degree sequences. Annals of Combinatorics 6: 125–145.CrossRefGoogle Scholar
5.Chung, F. & Lu, L (2002). The average distances in random graphs with given expected degrees. Proceedings of the National Academy of Sciences, USA 99: 15,879–15,882.CrossRefGoogle ScholarPubMed
6.Deijfen, M., van den Esker, H., van den Hofstad, R., & Hooghiemstra, G. (2009). A preferential attachment model with random initial degrees. Arkiv för Matematik 47: 41–72.CrossRefGoogle Scholar
7.Dorogovtsev, S. & Mendes, J. (2003). Evolution of networks, from biological nets to the Internet and WWW. Oxford: Oxford University Press.CrossRefGoogle Scholar
8.Fill, J., Scheinerman, E. & Singer-Cohen, K. (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. & Jaworski, J. (2002). Two models of random intersection graphs for classification. In Schwaiger, M. & Opitz, O. (eds.) Exploratory data analysis in empirical research. New York: Springer, pp. 67–81.Google Scholar
10.Jaworski, J., Karoński, M., & Stark, D. (2006). The degree of a typical vertex in generalized random intersection graph models. Discrete Mathematics 306: 2152–2165.CrossRefGoogle Scholar
11.Karoński, M., Scheinerman, E., & Singer-Cohen, K. (1999). On random intersection graphs: The subgraphs problem. Combinatorics, Probability & Computing 8: 131–159.CrossRefGoogle Scholar
12.Molloy, M., & Reed, B. (1995). A critical point for random graphs with a given degree sequence. Random Structures & Algorithms 6: 161–179.CrossRefGoogle Scholar
13.Molloy, M., & Reed, B. (1998). The size of the giant component of a random graph with a given degree sequence. Combinatorics, Probability & Computing 7: 295–305.CrossRefGoogle Scholar
14.Newman, M. E. J. (2003). Properties of highly clustered networks. Physical Review E 68: 026121.CrossRefGoogle ScholarPubMed
15.Newman, M. E. J. & Park, J. (2003). Why social networks are different from other types of networks. Physical Review E 68: 036122.CrossRefGoogle ScholarPubMed
16.Newman, M. E. J., Strogatz, S. H., & Watts, D. J. (2002). Random graphs with arbitrary degree distributions and their applications. Physical Review E 64: 026118.CrossRefGoogle Scholar
17.Palla, G., Derényi, I., Farkas, I., & Vicsek, T. (2005). Uncovering the overlapping community structure of complex networks in nature and society. Nature 435: 814–818.CrossRefGoogle ScholarPubMed
18.Singer, K. (1995). Random intersection graphs, Ph.D. thesis, Johns Hopkins University. Baltimore, MD.Google Scholar
19.Stark, D. (2004). The vertex degree distribution of random intersection graphs. Random Structures & Algorithms 24: 249–258.CrossRefGoogle Scholar
20.Yao, X., Zhang, C., Chen, J., & Li, Y. (2005). On the scale-free intersection graphs. Lecture Notes in Computer Science No. 3481. Berlin: Springer, pp. 1217–1224.Google Scholar