Skip to main content
SpringerLink
Log in
Book cover

Data Science, Learning by Latent Structures, and Knowledge Discovery pp 69–78Cite as

Recent Progress in Complex Network Analysis: Models of Random Intersection Graphs

  • Conference paper

Abstract

Experimental results show that in large complex networks such as Internet or biological networks, there is a tendency to connect elements which have a common neighbor. This tendency in theoretical random graph models is depicted by the asymptotically constant clustering coefficient. Moreover complex networks have power law degree distribution and small diameter (small world phenomena), thus these are desirable features of random graphs used for modeling real life networks. We survey various variants of random intersection graph models, which are important for networks modeling.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  • Barbour, A. D., & Reinert, G. (2011). The shortest distance in random multi-type intersection graphs. Random Structures and Algorithms, 39, 179–209.

    Article  MATH  MathSciNet  Google Scholar 

  • Behrisch, M. (2007). Component evolution in random intersection graphs. The Electronic Journal of Combinatorics, 14(1).

    Google Scholar 

  • Blackburn, S., & Gerke, S. (2009). Connectivity of the uniform random intersection graph. Discrete Mathematics, 309, 5130–5140.

    Article  MATH  MathSciNet  Google Scholar 

  • Bloznelis, M. (2008). Degree distribution of a typical vertex in a general random intersection graph. Lithuanian Mathematical Journal, 48, 38–45.

    Article  MATH  MathSciNet  Google Scholar 

  • Bloznelis, M. (2010a). A random intersection digraph: Indegree and outdegree distributions. Discrete Mathematics, 310, 2560–2566.

    Article  MATH  MathSciNet  Google Scholar 

  • Bloznelis, M. (2010b). Component evolution in general random intersection graphs. SIAM Journal on Discrete Mathematics, 24, 639–654.

    Article  MATH  MathSciNet  Google Scholar 

  • Bloznelis, M. (2010c). The largest component in an inhomogeneous random intersection graph with clustering. The Electronic Journal of Combinatorics, 17(1), R110.

    MathSciNet  Google Scholar 

  • Bloznelis, M. (2013). Degree and clustering coefficient in sparse random intersection graphs. The Annals of Applied Probability, 23, 1254–1289.

    Article  MATH  MathSciNet  Google Scholar 

  • Bloznelis, M., & Damarackas, J. (2013). Degree distribution of an inhomogeneous random intersection graph. The Electronic Journal of Combinatorics, 20(3), R3.

    MathSciNet  Google Scholar 

  • Bloznelis, M., Godehardt, E., Jaworski, J., Kurauskas, V., & Rybarczyk, K. (2015). Recent progress in complex network analysis: Properties of random intersection graphs. In B. Lausen, S. Krolak-Schwerdt, & M. Boehmer (Eds.), European Conference on Data Analysis. Berlin/Heidelberg/New York: Springer.

    Google Scholar 

  • Bloznelis, M., Jaworski, J., & Kurauskas, V. (2013). Assortativity and clustering of sparse random intersection graphs. Electronic Journal of Probability, 18, N-38.

    Google Scholar 

  • Bloznelis, M., Jaworski, J., & Rybarczyk, K. (2009). Component evolution in a secure wireless sensor network. Networks, 53(1), 19–26.

    Article  MATH  MathSciNet  Google Scholar 

  • Bloznelis, M., & Karoński, M. (2013). Random intersection graph process. In A. Bonato, M. Mitzenmacher, & P. Pralat (Eds.), Algorithms and models for the web graph. WAW 2013. Lecture notes in computer science (Vol. 8305, pp. 93–105). Switzerland: Springer International Publishing.

    Google Scholar 

  • Bloznelis, M., & Kurauskas, V. (2012). Clustering function: A measure of social influence. http://www.arxiv.org/abs/1207.4941.

  • Bloznelis, M., & Łuczak, T. (2013). Perfect matchings in random intersection graphs. Acta Mathematica Hungarica, 138, 15–33.

    Article  MATH  MathSciNet  Google Scholar 

  • Britton, T., Deijfen, M., Lindholm, M., & Lagerås, N. A. (2008). Epidemics on random graphs with tunable clustering. Journal of Applied Probability, 45, 743–756.

    Article  MATH  MathSciNet  Google Scholar 

  • Deijfen, M., & Kets, W. (2009). Random intersection graphs with tunable degree distribution and clustering. Probability in the Engineering and Informational Sciences, 23, 661–674.

    Article  MATH  MathSciNet  Google Scholar 

  • Eschenauer, L., & Gligor, V. D. (2002). A key-management scheme for distributed sensor networks. In Proceedings of the 9th ACM Conference on Computer and Communications Security (pp. 41–47).

    Google Scholar 

  • Fill, J. A., Scheinerman, E. R., & 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 and Algorithms, 16, 156–176.

    Google Scholar 

  • Godehardt, E., & Jaworski, J. (2001). Two models of random intersection graphs and their applications. Electronic Notes in Discrete Mathematics, 10, 129–132.

    Article  MathSciNet  Google Scholar 

  • Godehardt, E., & Jaworski, J. (2003). Two models of random intersection graphs for classification. In M. Schwaiger & O. Opitz (Eds.), Exploratory data analysis in empirical research (pp. 67–81). Berlin/Heidelberg/New York: Springer.

    Chapter  Google Scholar 

  • Godehardt, E., Jaworski, J., & Rybarczyk, K. (2007). Random intersection graphs and classification. In R. Decker & H.-J. Lenz (Eds.), Advances in data analysis (pp. 67–74). Berlin/Heidelberg/New York: Springer.

    Chapter  Google Scholar 

  • Godehardt, E., Jaworski, J., & Rybarczyk, K. (2012). Clustering coefficients of random intersection graphs. In W. Gaul, A. Geier-Schulz, L. Schmidt-Thieme, & J. Kunze (Eds.), Challenges at the interface of data analysis, computer science, and optimization (pp. 243–253). Berlin/Heidelberg/New York: Springer.

    Chapter  Google Scholar 

  • Guillaume, J. L., & Latapy, M. (2004). Bipartite structure of all complex networks. Information Processing Letters, 90, 215–221.

    Article  MATH  MathSciNet  Google Scholar 

  • Jaworski, J., Karoński, M., & Stark, D. (2006). The degree of a typical vertex in generalized random intersection graph models. Discrete Mathematics, 306, 2152–2165.

    Article  MATH  MathSciNet  Google Scholar 

  • Jaworski, J., & Stark, D. (2008). The vertex degree distribution of passive random intersection graph models. Combinatorics, Probability and Computing, 17, 549–558.

    Article  MATH  MathSciNet  Google Scholar 

  • Johnson, J. R., & Markström, K. (2013). Turán and Ramsey properties of subcube intersection graphs. Combinatorics, Probability and Computing, 22(1), 55–70.

    Article  MATH  MathSciNet  Google Scholar 

  • Karoński, M., Scheinerman, E. R., & Singer-Cohen, K. B. (1999). On random intersection graphs: The subgraph problem. Combinatorics, Probability and Computing, 8, 131–159.

    Article  MATH  MathSciNet  Google Scholar 

  • Lagerås, A. N., & Lindholm, M. (2008). A note on the component structure in random intersection graphs with tunable clustering. Electronic Journal of Combinatorics, 15(1).

    Google Scholar 

  • Martin, T., Ball, B., Karrer, B., & Newman, M. E. J. (2013). Coauthorship and citation patterns in the Physical Review. Phys. Rev. E 88, 012814.

    Article  Google Scholar 

  • Newman, M. E. J., Watts, D. J., & Strogatz, S. H. (2002). Random graph models of social networks. Proceedings of the National Academy of Sciences of the USA, 99(Suppl. 1), 2566–2572.

    Article  MATH  Google Scholar 

  • Nikoletseas, S., Raptopoulos, C., & Spirakis, P. (2004). The existence and efficient construction of large independent sets in general random intersection graphs. In J. Daz, J. Karhumki, A. Lepist, & D. Sannella (Eds.), ICALP. Lecture notes in computer science (Vol. 3142, pp. 1029–1040). Berlin: Springer.

    Google Scholar 

  • Nikoletseas, S., Raptopoulos, C., & Spirakis, P. (2008). Large independent sets in general random intersection graphs. Theoretical Computer Science, 406, 215–224.

    Article  MATH  MathSciNet  Google Scholar 

  • Nikoletseas, S., Raptopoulos, C., & Spirakis, P. G. (2011). On the independence number and Hamiltonicity of uniform random intersection graphs. Theoretical Computer Science, 412, 6750–6760.

    Article  MATH  MathSciNet  Google Scholar 

  • Rybarczyk, K. (2011a). Diameter, connectivity, and phase transition of the uniform random intersection graph. Discrete Mathematics, 311, 1998–2019.

    Article  MATH  MathSciNet  Google Scholar 

  • Rybarczyk, K. (2011b). Equivalence of the random intersection graph and G(n, p). Random Structures and Algorithms, 38, 205–234.

    Google Scholar 

  • Rybarczyk, K. (2011c). Sharp threshold functions for random intersection graphs via a coupling method. The Electronic Journal of Combinatorics, 18(1), P36.

    MathSciNet  Google Scholar 

  • Rybarczyk, K. (2012). The degree distribution in random intersection graphs. In W. Gaul, A. Geier-Schulz, L. Schmidt-Thieme, & J. Kunze (Eds.), Challenges at the interface of data analysis, computer science, and optimization (pp. 291–299). Berlin/Heidelberg/New York: Springer.

    Chapter  Google Scholar 

  • Rybarczyk, K. (2013). The coupling method for inhomogeneous random intersection graphs. ArXiv:1301.0466.

    Google Scholar 

  • Shang, Y. (2010). Degree distributions in general random intersection graphs. The Electronical Journal of Combinatorics, 17, #R23.

    Google Scholar 

  • Stark, D. (2004). The vertex degree distribution of random intersection graphs. Random Structures and Algorithms, 24, 249–258.

    Article  MATH  MathSciNet  Google Scholar 

  • Strogatz, S. H., & Watts, D. J. (1998). Collective dynamics of small-world networks. Nature, 393, 440–442.

    Article  Google Scholar 

  • Yagan, O., & Makowski, A. M. (2009). Random key graphs – Can they be small worlds? In 2009 First International Conference on Networks & Communications (pp. 313–318).

    Google Scholar 

Download references

Acknowledgements

The work of M. Bloznelis and V. Kurauskas was supported by the Lithuanian Research Council (grant MIP–067/2013). J. Jaworski and K. Rybarczyk were supported by the National Science Centre—DEC-2011/01/B/ST1/03943. Co-operation between E. Godehardt and J. Jaworski was also supported by Deutsche Forschungsgemeinschaft (grant no. GO 490/17–1).

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Informatics, Vilnius University, 03225, Vilnius, Lithuania

    Mindaugas Bloznelis & Valentas Kurauskas

  2. Clinic of Cardiovascular Surgery, Heinrich Heine University, 40225, Düsseldorf, Germany

    Erhard Godehardt

  3. Faculty of Mathematics and Computer Science, Adam Mickiewicz University, 60769, Poznań, Poland

    Jerzy Jaworski & Katarzyna Rybarczyk

Authors
  1. Mindaugas Bloznelis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Erhard Godehardt
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jerzy Jaworski
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Valentas Kurauskas
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Katarzyna Rybarczyk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Erhard Godehardt .

Editor information

Editors and Affiliations

  1. University of Essex, Colchester, United Kingdom

    Berthold Lausen

  2. University of Luxembourg, Walferdange, Luxembourg

    Sabine Krolak-Schwerdt

  3. University of Luxembourg, Walferdange, Luxembourg

    Matthias Böhmer

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bloznelis, M., Godehardt, E., Jaworski, J., Kurauskas, V., Rybarczyk, K. (2015). Recent Progress in Complex Network Analysis: Models of Random Intersection Graphs. In: Lausen, B., Krolak-Schwerdt, S., Böhmer, M. (eds) Data Science, Learning by Latent Structures, and Knowledge Discovery. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44983-7_6

Download citation

Publish with us

Policies and ethics

Search

Navigation