The degree-restricted random process is far from uniform

TL;DR

This paper demonstrates that the degree-restricted random process significantly diverges from uniform randomness for non-regular degree sequences, using a novel combinatorial switching method adapted for stochastic process analysis.

Contribution

It introduces a new combinatorial switching technique to analyze the degree-restricted process, showing its deviation from uniform models for non-regular sequences.

Findings

Degree-restricted process differs from uniform random graphs for non-regular sequences.

The switching method is effectively adapted to stochastic processes.

The process aligns with uniform models only in nearly regular cases.

Abstract

The degree-restricted random process is a natural algorithmic model for generating graphs with degree sequence D_n=(d_1, \ldots, d_n): starting with an empty n-vertex graph, it sequentially adds new random edges so that the degree of each vertex v_i remains at most d_i. Wormald conjectured in 1999 that, for d-regular degree sequences D_n, the final graph of this process is similar to a uniform random d-regular graph. In this paper we show that, for degree sequences D_n that are not nearly regular, the final graph of the degree-restricted random process differs substantially from a uniform random graph with degree sequence D_n. The combinatorial proof technique is our main conceptual contribution: we adapt the switching method to the degree-restricted process, demonstrating that this enumeration technique can also be used to analyze stochastic processes (rather than just uniform random models, as before).