On the swap-distances of different realizations of a graphical degree sequence

TL;DR

This paper develops formulas for the swap-distance between different realizations of a degree sequence, improving bounds and aiding in sampling and analysis of social networks.

Contribution

It introduces Gallai-type identities that precisely compute swap-distances, enhancing previous crude bounds for simple and directed graphs.

Findings

Derived formulas for swap-distances between realizations.

Significantly improved upper bounds on swap-distances.

Applicable to both undirected and directed degree sequences.

Abstract

One of the first graph theoretical problems which got serious attention (already in the fifties of the last century) was to decide whether a given integer sequence is equal to the degree sequence of a simple graph (or it is {\em graphical} for short). One method to solve this problem is the greedy algorithm of Havel and Hakimi, which is based on the {\em swap} operation. Another, closely related question is to find a sequence of swap operations to transform one graphical realization into another one of the same degree sequence. This latter problem got particular emphases in connection of fast mixing Markov chain approaches to sample uniformly all possible realizations of a given degree sequence. (This becomes a matter of interest in connection of -- among others -- the study of large social networks.) Earlier there were only crude upper bounds on the shortest possible length of such swap sequences between two realizations. In this paper we develop formulae (Gallai-type identities) for these {\em swap-distance}s of any two realizations of simple undirected or directed degree sequences. These identities improves considerably the known upper bounds on the swap-distances.