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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)

Abstract

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.

Keywords

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.

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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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