On Realizations of a Joint Degree Matrix

TL;DR

This paper proves the connectivity of all graph realizations with a given joint degree matrix via restricted swaps, corrects previous proof errors, and discusses implications for sampling methods.

Contribution

It provides a corrected, simplified proof of the connectivity of graph realizations with a fixed joint degree matrix and explores sampling implications.

Findings

Connectivity of all realizations established

Simplified proof of joint degree matrix conditions

Discussion on mixing time of sampling algorithms

Abstract

The joint degree matrix of a graph gives the number of edges between vertices of degree i and degree j for every pair (i,j). One can perform restricted swap operations to transform a graph into another with the same joint degree matrix. We prove that the space of all realizations of a given joint degree matrix over a fixed vertex set is connected via these restricted swap operations. This was claimed before, but there is an error in the previous proof, which we illustrate by example. We also give a simplified proof of the necessary and sufficient conditions for a matrix to be a joint degree matrix. Finally, we address some of the issues concerning the mixing time of the corresponding MCMC method to sample uniformly from these realizations.