# On Realizations of a Joint Degree Matrix

arXiv: https://arxiv.org/abs/1302.3548 · HTML: https://mlsanity.com/paper/1302.3548/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Feb 2013.

## Built on by (most cited first, 1 of 1)

- [Alice and the Caterpillar: A more descriptive null model for assessing data mining results](https://mlsanity.com/paper/2506.09764/) (2025)

## Abstract (arXiv)

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.