{
  "abstract": "We study post-hoc refinement of frozen node classifiers: given only the graph $G$ and class distributions $Q$ predicted by a frozen model, can we improve accuracy without access to node features, model parameters, or gradients? APPNP answers this by propagating logits with a restart towards the initial predictions, minimizing the anchored Dirichlet energy. Instead, we consider the Potts energy, and decompose it into a Dirichlet term, which penalizes disagreement between neighbouring nodes, and a Gini term, which penalizes indecision within each node. This decomposition motivates Propagate, Then Sharpen (PtS), which alternates between propagation of class probabilities and node-wise, mass-preserving sharpening, with only one additional hyperparameter selected using labelled validation nodes. Across nine homophilic graphs, with a frozen MLP backbone, PtS improves mean test accuracy over independently tuned APPNP by $1.71$ percentage points on clean inputs and $3.90$ under severe Gaussian feature corruption. Gains over APPNP become smaller, but remain positive with frozen GCN and GraphSAGE backbones. Sharpening also removes most of the accuracy loss of deep propagation: on clean inputs without restart, accuracy falls by $2.2$ points between $2$ and $100$ propagation steps under PtS, compared with $33.8$ for APPNP.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35080",
  "author_total": 3,
  "authors": [
    {
      "name": "Preben Johnsen Bentdal"
    },
    {
      "name": "Nello Blaser"
    },
    {
      "name": "Xue-Cheng Tai"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e7n2403jmt9qk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35080",
  "fulltext_url": null,
  "id": "e7n2403jmt9qk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35080",
  "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.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Advanced Graph Neural Networks"
    ]
  ],
  "openalex": "W7214751092",
  "pdf_url": "https://arxiv.org/pdf/2609.35080",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e7n2403jmt9qk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:26Z",
      "native_id": "2609.35080",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35080"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Propagate, Then Sharpen: Post-Hoc Refinement of Frozen Node Classifiers",
  "title_text": "Propagate, Then Sharpen: Post-Hoc Refinement of Frozen Node Classifiers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:26Z",
  "url": "https://research.draconitic.com/paper/e7n2403jmt9qk",
  "venue": "arXiv",
  "versions": []
}