Partitioning and modularity of graphs with arbitrary degree distribution
Disordered Systems and Neural Networks
2008-12-11 v2 Statistical Mechanics
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.
Cite
@article{arxiv.cond-mat/0606295,
title = {Partitioning and modularity of graphs with arbitrary degree distribution},
author = {Joerg Reichardt and Stefan Bornholdt},
journal= {arXiv preprint arXiv:cond-mat/0606295},
year = {2008}
}
Comments
Revised version including new plots and improved discussion of some mathematical details