Degree Stability of Parallel Greedy Matching in Random Graphs

Source abstract

We study a natural synchronous greedy matching process in Erdős-Rényi random graphs $\mathbb{G}(n,p)$ and give a rigorous analysis of this parallel stochastic dynamics. The process evolves as follows: in each round, every unmatched vertex proposes to a uniformly random neighbour, and mutually proposing pairs are matched and removed from the graph, after which the process repeats on the residual graph. We consider the density regime $p=Θ(\log^{1+ε} n/n)$ for any constant $ε>0$, which lies just above the threshold for the existence of a perfect matching. We show that, with high probability, for $Θ(\log^{1+ε} n)$ rounds, every unmatched vertex keeps a constant fraction of its initial degree in the residual graph, simultaneously for all vertices and all rounds. By then, the process has matched a constant fraction of the vertices. Our analysis tracks fine-grained invariants that control the evolution of individual vertex degrees, rather than directly tracking the global degree distribution. At the local level, we analyze the per-round degree loss of a vertex via a neighbourhood-exposure martingale. We show that structural sparsity of $\mathbb{G}(n,p)$ yields $O(1)$ variance and thus the per-round degree loss is strongly concentrated around its conditional expectation. At the global level, these local concentration bounds control the increments of a martingale tracking cumulative degree loss over many rounds, allowing us to show that vertex degrees remain stable for $Ω(\log^{1+ε} n)$ rounds.