Partitioning and modularity of graphs with arbitrary degree distribution

TL;DR

This paper analyzes the graph bi-partitioning problem in dense graphs with arbitrary degree distributions, revealing a universal scaling law for cut-size and extending to q-partitioning, with implications for community detection.

Contribution

It introduces a replica method-based solution for bi-partitioning in dense graphs with arbitrary degree distributions, generalizing previous Poissonian results and enabling modularity assessment.

Findings

Cut-size scales universally with <k^1/2> in dense graphs.

Results extend to q-partitioning problems.

Allows assessment of statistical significance in community detection.

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 <k^1/2>. In contrast, earlier results studying the problem in graphs with a Poissonian degree distribution had found a scaling with <k>^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.