{
  "abstract": "We study a semidefinite programming (SDP) relaxation for community recovery when the number of communities grows logarithmically. In the balanced stochastic block model with $n=km$ vertices, we consider the regime $k/\\log m\\toγ>0$, with edge probabilities $α\\log m/m$ within communities and $β\\log m/m$ across them, for fixed $α>β>0$. We derive the sharp asymptotic tightness boundary away from critical cases. When rare vertices cause tightness to fail while the bulk remains spectrally stable, the normalized matrix error of every near-optimal solution still vanishes. We prove matching high-probability exponents for this error and the normalized optimal objective gain, governed by the same local correction that determines tightness. Throughout this spectrally stable region, a single SDP solve followed by explicit rounding and refinement achieves exact community recovery above the information-theoretic threshold. This guarantee holds even when the planted community matrix is not an optimal solution to the SDP.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09303",
  "author_total": 3,
  "authors": [
    {
      "name": "Shengtao Guo"
    },
    {
      "name": "Ethan X. Fang"
    },
    {
      "name": "Junwei Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1cf0t74fb3tsa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09303",
  "fulltext_url": null,
  "id": "1cf0t74fb3tsa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09303",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "3109|Complex Network Analysis Techniques"
    ]
  ],
  "openalex": "W7220961156",
  "pdf_url": "https://arxiv.org/pdf/2610.09303",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1cf0t74fb3tsa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.09303",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09303"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Tightness and Error Exponents of SDP with Logarithmically Many Communities",
  "title_text": "Tightness and Error Exponents of SDP with Logarithmically Many Communities",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:48Z",
  "url": "https://research.draconitic.com/paper/1cf0t74fb3tsa",
  "venue": "arXiv",
  "versions": []
}