The degree-restricted random process is far from uniform

Abstract

The degree-restricted random process is a natural algorithmic model for generating graphs with degree sequence $d_n=(d_1,\dots ,d_n)$: 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).