Degree Stability of Parallel Greedy Matching in Random Graphs
Abstract
We study a natural synchronous greedy matching process in Erdős-Rényi random graphs 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(log1+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(log1+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 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(log1+ε n) rounds.
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BibTeX
@misc{vishvajeet2026degree,
title = {{Degree Stability of Parallel Greedy Matching in Random Graphs}},
author = {Vishvajeet N and Shrinidhi Teganahally Sridhara},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25507},
eprint = {2610.25507},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25507},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Degree Stability of Parallel Greedy Matching in Random Graphs
AU - Vishvajeet N
AU - Shrinidhi Teganahally Sridhara
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25507
UR - https://arxiv.org/abs/2610.25507
AB - We study a natural synchronous greedy matching process in Erdős-Rényi random graphs 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 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.
ER -