{
  "abstract": "Graph learning from edge-detecting queries concerns the reconstruction of an unknown edge set on a known vertex set. Each query reports whether a specified vertex subset contains at least one edge. We study non-adaptive schemes, in which all queries are fixed before any outcomes are observed, with the goal of achieving exact recovery using few queries and fast decoding. For general graphs on $n$ vertices with at most $k$ edges, non-adaptive recovery requires $Ω(\\min\\{k^2\\log n,n^2\\})$ queries in the worst case, even when a small error probability is allowed. In this paper, we consider Erdős--Rényi ($\\mathrm{ER}$) graphs $G\\sim \\mathrm{ER}(n,q)$, with expected edge count $\\bar{k}=q\\binom{n}{2}$. Our scheme uses $O(\\bar{k}\\log n)$ queries and achieves exact recovery in $O(\\bar{k}\\log n)$ decoding time with probability tending to one throughout the regime $\\bar{k}\\to\\infty$ and $\\bar{k}=o(n^2)$. This improves the previous $O(\\bar{k}^{1+δ}\\log n)$ decoding guarantee for any fixed $δ>0$, while maintaining the same query order. The guarantee also extends beyond the previously studied regime $\\bar{k}=Θ(n^{2θ})$ with fixed $θ\\in(0,1)$. Our approach builds on the binary splitting method used in prior work, which organizes vertices into a hierarchy of successively smaller groups. We introduce three main changes: (i) we use random affine hash functions over a finite field to process each candidate pair in constant time; (ii) we apply the splitting procedure directly to the full graph, avoiding the need to combine solutions to multiple smaller graph-learning subproblems; and (iii) we bound the total decoding workload directly rather than deriving separate high-probability bounds on candidate counts at each level.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33704",
  "author_total": 1,
  "authors": [
    {
      "name": "Hoang Ta"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1tq1tts3656ja.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33704",
  "fulltext_url": null,
  "id": "1tq1tts3656ja",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33704",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7214773514",
  "pdf_url": "https://arxiv.org/pdf/2609.33704",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1tq1tts3656ja.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33704",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33704"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Non-Adaptive Learning of Sparse Erdős--Rényi Graphs via Affine Splitting",
  "title_text": "Non-Adaptive Learning of Sparse Erdős--Rényi Graphs via Affine Splitting",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/1tq1tts3656ja",
  "venue": "arXiv",
  "versions": []
}