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.Comment: Revised version including new plots and improved discussion of some
mathematical detail