Uniformly sampling graphs with self-loops and a given degree sequence

Joel Nishimura

Published 2017-01-17Version 1

`Double edge swaps' transform one graph into another while preserving the graph's degree sequence, and have thus been used in a number of popular Markov chain Monte Carlo (MCMC) sampling techniques. However, while double edge-swap MCMC sampling can, for any fixed degree sequence, sample simple graphs, multigraphs, and pseudographs uniformly, this is not true for graphs which allow self-loops but not multiedges (loopy graphs). Indeed, we exactly characterize the degree sequences where double edge swaps cannot reach every valid loopy graph and develop an efficient algorithm to determine such degree sequences. The same classification scheme to characterize degree sequences can be used to prove that, for all degree sequences, loopy graphs are connected by a combination of double and triple edge swaps. Thus, we contribute the first MCMC sampler that uniformly samples loopy graphs with any given sequence.