Generating maximally disassortative graphs with given degree distribution

Abstract

In this paper we consider the optimization problem of generating graphs with a prescribed degree distribution, such that the correlation between the degrees of connected nodes, as measured by Spearman's rho, is minimal. We provide an algorithm for solving this problem and obtain a complete characterization of the joint degree distribution in these maximally disassortative graphs, in terms of the size-biased degree distribution. As a result we get a lower bound for Spearman's rho on graphs with an arbitrary given degree distribution. We use this lower bound to show that for any fixed tail exponent, there exist scale-free degree sequences with this exponent such that the minimum value of Spearman's rho for all graphs with such degree sequences is arbitrary close to zero. This implies that specifying only the tail behavior of the degree distribution, as is often done in the analysis of complex networks, gives no guarantees for the minimum value of Spearman's rho.