Correlations in connected random graphs

TL;DR

This paper analyzes degree correlations in connected random graphs, deriving formulas for joint degree distributions, revealing disassortativity, and providing an efficient graph generation algorithm.

Contribution

It introduces analytic formulas for degree correlations in connected random graphs and an efficient algorithm for generating such graphs.

Findings

Connected graphs are disassortative.

Degree correlations are strongly influenced by leaves.

Analytic formulas for joint degree distributions are derived.

Abstract

We study the properties of the giant connected component in random graphs with arbitrary degree distribution. We concentrate on the degree-degree correlations. We show that the adjoining nodes in the giant connected component are correlated and derive analytic formulas for the joint nearest-neighbor degree probability distribution. Using those results we describe the correlations in maximal entropy connected random graphs. We show that connected graphs are disassortative and that correlations are strongly related to the presence of one-degree nodes (leaves). We propose an efficient algorithm for generating connected random graphs. We illustrate our results with several examples.