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Partitioning and modularity of graphs with arbitrary degree distribution

Joerg Reichardt·Stefan Bornholdt
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Abstract

We solve the graph bi-partitioning problem in dense graphs with arbitrary degree distribution using the replica method. We find the cut-size to scale universally with . In contrast, earlier results studying the problem in graphs with a Poissonian degree distribution had found a scaling with ^1/2 [Fu and Anderson, J. Phys. A: Math. Gen. 19, 1986]. The new results also generalize to the problem of q-partitioning. They can be used to find the expected modularity Q [Newman and Grivan, Phys. Rev. E, 69, 2004] of random graphs and allow for the assessment of statistical significance of the output of community detection algorithms.

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Joerg Reichardt, Stefan Bornholdt. 2007-04-06. Partitioning and modularity of graphs with arbitrary degree distribution. https://doi.org/10.1103/physreve.76.015102

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