## The Annals of Applied Probability

### Degree and clustering coefficient in sparse random intersection graphs

Mindaugas Bloznelis

#### Abstract

We establish asymptotic vertex degree distribution and examine its relation to the clustering coefficient in two popular random intersection graph models of Godehardt and Jaworski [Electron. Notes Discrete Math. 10 (2001) 129–132]. For sparse graphs with a positive clustering coefficient, we examine statistical dependence between the (local) clustering coefficient and the degree. Our results are mathematically rigorous. They are consistent with the empirical observation of Foudalis et al. [In Algorithms and Models for Web Graph (2011) Springer] that, “clustering correlates negatively with degree.” Moreover, they explain empirical results on $k^{-1}$ scaling of the local clustering coefficient of a vertex of degree $k$ reported in Ravasz and Barabási [Phys. Rev. E 67 (2003) 026112].

#### Article information

Source
Ann. Appl. Probab., Volume 23, Number 3 (2013), 1254-1289.

Dates
First available in Project Euclid: 7 March 2013

https://projecteuclid.org/euclid.aoap/1362684860

Digital Object Identifier
doi:10.1214/12-AAP874

Mathematical Reviews number (MathSciNet)
MR3076684

Zentralblatt MATH identifier
1273.05197

Subjects

#### Citation

Bloznelis, Mindaugas. Degree and clustering coefficient in sparse random intersection graphs. Ann. Appl. Probab. 23 (2013), no. 3, 1254--1289. doi:10.1214/12-AAP874. https://projecteuclid.org/euclid.aoap/1362684860

#### References

• [1] Barbour, A. D. and Reinert, G. (2011). The shortest distance in random multi-type intersection graphs. Random Structures Algorithms 39 179–209.
• [2] Barrat, A. and Weigt, M. (2000). On the properties of small-world networks. Eur. Phys. J. B 13 547–560.
• [3] Blackburn, S. R. and Gerke, S. (2009). Connectivity of the uniform random intersection graph. Discrete Math. 309 5130–5140.
• [4] Bloznelis, M. (2008). Degree distribution of a typical vertex in a general random intersection graph. Lith. Math. J. 48 38–45.
• [5] Bloznelis, M. (2010). A random intersection digraph: Indegree and outdegree distributions. Discrete Math. 310 2560–2566.
• [6] Bloznelis, M. (2010). The largest component in an inhomogeneous random intersection graph with clustering. Electron. J. Combin. 17 Research Paper 110, 17.
• [8] Britton, T., Deijfen, M., Lagerås, A. N. and Lindholm, M. (2008). Epidemics on random graphs with tunable clustering. J. Appl. Probab. 45 743–756.
• [9] Deijfen, M. and Kets, W. (2009). Random intersection graphs with tunable degree distribution and clustering. Probab. Engrg. Inform. Sci. 23 661–674.
• [10] Eschenauer, L. and Gligor, V. D. (2002). A key-management scheme for distributed sensor networks. In Proceedings of the 9th ACM Conference on Computer and Communications Security, Washington, DC 41–47.
• [11] Foss, S., Korshunov, D. and Zachary, S. (2011). An Introduction to Heavy-Tailed and Subexponential Distributions. ACM, New York.
• [12] Foudalis, I., Jain, K., Papadimitriou, C. and Sideri, M. (2011). Modeling social networks through user background and behavior. In Algorithms and Models for the Web Graph. Lecture Notes in Computer Science 6732 85–102. Springer, Heidelberg.
• [13] Godehardt, E. and Jaworski, J. (2001). Two models of random intersection graphs and their applications. Electron. Notes Discrete Math. 10 129–132.
• [14] Godehardt, E. and Jaworski, J. (2003). Two models of random intersection graphs for classification. In Exploratory Data Analysis in Empirical Research 67–81. Springer, Berlin.
• [15] Godehardt, E., Jaworski, J. and Rybarczyk, K. (2012). Clustering coefficients of random intersection graphs. In Studies in Classification, Data Analysis and Knowledge Organization 243–253. Springer, Berlin.
• [16] Guillaume, J.-L. and Latapy, M. (2004). Bipartite structure of all complex networks. Inform. Process. Lett. 90 215–221.
• [17] Jaworski, J., Karoński, M. and Stark, D. (2006). The degree of a typical vertex in generalized random intersection graph models. Discrete Math. 306 2152–2165.
• [18] Jaworski, J. and Stark, D. (2008). The vertex degree distribution of passive random intersection graph models. Combin. Probab. Comput. 17 549–558.
• [19] Karoński, M., Scheinerman, E. R. and Singer-Cohen, K. B. (1999). On random intersection graphs: The subgraph problem. Combin. Probab. Comput. 8 131–159.
• [20] Newman, M. E. J. (2003). Properties of highly clustered networks. Phys. Rev. E 68 026121.
• [21] 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.
• [22] Newman, M. E. J., Watts, D. J. and Strogatz, S. H. (2002). Random graph models of social networks. Proc. Natl. Acad. Sci. USA 99 (Suppl. 1) 2566–2572.
• [23] Nikoletseas, S., Raptopoulos, C. and Spirakis, P. G. (2011). On the independence number and Hamiltonicity of uniform random intersection graphs. Theoret. Comput. Sci. 412 6750–6760.
• [24] Ravasz, L. and Barabási, A. L. (2003). Hierarchical organization in complex networks. Phys. Rev. E 67 026112.
• [25] Rybarczyk, K. (2011). Diameter, connectivity, and phase transition of the uniform random intersection graph. Discrete Math. 311 1998–2019.
• [26] Rybarczyk, K. (2012). The degree distribution in random intersection graphs. In Challenges at the Interface of Data Analysis, Computer Science, and Optimization 291–299. Springer, Berlin.
• [27] Stark, D. (2004). The vertex degree distribution of random intersection graphs. Random Structures Algorithms 24 249–258.
• [28] Steele, J. M. (1994). Le Cam’s inequality and Poisson approximations. Amer. Math. Monthly 101 48–54.
• [29] Watts, D. J. and Strogatz, S. H. (1998). Collective dynamics of “small-world” networks. Nature 393 440–442.
• [30] Yagan, O. and Makowski, A. M. (2009). Random key graphs—Can they be small worlds? In NETCOM’09: Proceedings of the 2009 First International Conference on Networks & Communications 313–318. IEEE Computer Society, Washington, DC.