Realistic, Mathematically Tractable Graph Generation and Evolution, Using Kronecker Multiplication

  • Jurij Leskovec
  • Deepayan Chakrabarti
  • Jon Kleinberg
  • Christos Faloutsos
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3721)


How can we generate realistic graphs? In addition, how can we do so with a mathematically tractable model that makes it feasible to analyze their properties rigorously? Real graphs obey a long list of surprising properties: Heavy tails for the in- and out-degree distribution; heavy tails for the eigenvalues and eigenvectors; small diameters; and the recently discovered “Densification Power Law” (DPL). All published graph generators either fail to match several of the above properties, are very complicated to analyze mathematically, or both. Here we propose a graph generator that is mathematically tractable and matches this collection of properties. The main idea is to use a non-standard matrix operation, the Kronecker product, to generate graphs that we refer to as “Kronecker graphs”.

We show that Kronecker graphs naturally obey all the above properties; in fact, we can rigorously prove that they do so. We also provide empirical evidence showing that they can mimic very well several real graphs.


Random Graph Degree Distribution Kronecker Product Heavy Tail Scree Plot 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Albert, R., Barabasi, A.-L.: Emergence of scaling in random networks. Science, 509–512 (1999)Google Scholar
  2. 2.
    Albert, R., Barabási, A.-L.: Statistical mechanics of complex networks. Reviews of Modern Physics (2002)Google Scholar
  3. 3.
    Bak, P.: How nature works: The science of self-organized criticality (September 1996)Google Scholar
  4. 4.
    Barabasi, A.-L., Ravasz, E., Vicsek, T.: Deterministic scale-free networks. Physica A 299, 559–564 (2001)zbMATHCrossRefGoogle Scholar
  5. 5.
    Bi, Z., Faloutsos, C., Korn, F.: The DGX distribution for mining massive, skewed data. In: KDD, pp. 17–26 (2001)Google Scholar
  6. 6.
    Waxman, B.M.: Routing of multipoint connections. IEEE Journal on Selected Areas in Communications 6(9) (December 1988)Google Scholar
  7. 7.
    Broder, A., Kumar, R., Maghoul1, F., Raghavan, P., Rajagopalan, S., Stata, R., Tomkins, A., Wiener, J.: Graph structure in the web: experiments and models. In: Proceedings of World Wide Web Conference (2000)Google Scholar
  8. 8.
    Carlson, J.M., Doyle, J.: Highly optimized tolerance: a mechanism for power laws in designed systems. Physics Review E 60(2), 1412–1427 (1999)CrossRefGoogle Scholar
  9. 9.
    Chakrabarti, D., Zhan, Y., Faloutsos, C.: R-MAT: A recursive model for graph mining. In: SIAM Data Mining (2004)Google Scholar
  10. 10.
    Chow, T.: The Q-spectrum and spanning trees of tensor products of bipartite graphs. Proc. Amer. Math. Soc. 125, 3155–3161 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Erdos, P., Renyi, A.: On the evolution of random graphs. Publication of the Mathematical Institute of the Hungarian Acadamy of Science 5, 17–67 (1960)MathSciNetGoogle Scholar
  12. 12.
    Fabrikant, A., Koutsoupias, E., Papadimitriou, C.H.: Heuristically optimized trade-offs: A new paradigm for power laws in the internet (extended abstract) (2002)Google Scholar
  13. 13.
    Faloutsos, M., Faloutsos, P., Faloutsos, C.: On power-law relationships of the internet topology. In: SIGCOMM, pp. 251–262 (1999)Google Scholar
  14. 14.
    Girvan, M., Newman, M.E.J.: Community structure in social and biological networks. Proc. Natl. Acad. Sci. USA 99 (2002)Google Scholar
  15. 15.
    Kleinberg, J.M., Kumar, S.R., Raghavan, P., Rajagopalan, S., Tomkins, A.: The web as a graph: Measurements, models and methods. In: Proceedings of the International Conference on Combinatorics and Computing (1999)Google Scholar
  16. 16.
    Kumar, S.R., Raghavan, P., Rajagopalan, S., Tomkins, A.: Extracting large-scale knowledge bases from the web. In: VLDB, pp. 639–650 (1999)Google Scholar
  17. 17.
    Langville, A.N., Stewart, W.J.: The Kronecker product and stochastic automata networks. Journal of Computation and Applied Mathematics 167, 429–447 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Leskovec, J., Kleinberg, J., Faloutsos, C.: Graphs over time: densification laws, shrinking diameters and possible explanations. In: KDD 2005, Chicago, IL, USA (2005)Google Scholar
  19. 19.
    Loan, C.F.V.: The ubiquitous Kronecker product. Journal of Computation and Applied Mathematics 123, 85–100 (2000)zbMATHCrossRefGoogle Scholar
  20. 20.
    Mehta, M.: Random Matrices, 2nd edn. Academic Press, London (1991)zbMATHGoogle Scholar
  21. 21.
    Milgram, S.: The small-world problem. Psychology Today 2, 60–67 (1967)Google Scholar
  22. 22.
    Palmer, C.R., Gibbons, P.B., Faloutsos, C.: Anf: A fast and scalable tool for data mining in massive graphs. In: SIGKDD, Edmonton, AB, Canada (2002)Google Scholar
  23. 23.
    Pennock, D.M., Flake, G.W., Lawrence, S., Glover, E.J., Giles, C.L.: Winners don’t take all: Characterizing the competition for links on the web. Proceedings of the National Academy of Sciences 99(8), 5207–5211 (2002)zbMATHCrossRefGoogle Scholar
  24. 24.
    Redner, S.: How popular is your paper? an empirical study of the citation distribution. European Physical Journal B 4, 131–134 (1998)CrossRefGoogle Scholar
  25. 25.
    Sole, R., Goodwin, B.: Signs of Life: How Complexity Pervades Biology. Perseus Books Group, New York, NY (2000)Google Scholar
  26. 26.
    Tauro, S.L., Palmer, C., Siganos, G., Faloutsos, M.: A simple conceptual model for the internet topology. In: In Global Internet, San Antonio, Texas (2001)Google Scholar
  27. 27.
    Watts, D.J., Strogatz, S.H.: Collective dynamics of ’small-world’ networks. Nature 393, 440–442 (1998)CrossRefGoogle Scholar
  28. 28.
    Winick, J., Jamin, S.: Inet-3.0: Internet Topology Generator. Technical Report CSE-TR-456-02, University of Michigan, Ann Arbor (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Jurij Leskovec
    • 1
  • Deepayan Chakrabarti
    • 1
  • Jon Kleinberg
    • 2
  • Christos Faloutsos
    • 1
  1. 1.School of Computer ScienceCarnegie Mellon University 
  2. 2.Department of Computer ScienceCornell University 

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