Realistic, Mathematically Tractable Graph Generation and Evolution, Using Kronecker Multiplication
Jure Leskovec, Deepay Chakrabarti, Jon Kleinberg and Christos Faloutsos
In: ECML/PKDD 2005, Porto, Portugal(2005).

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.

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