{
  "schema_version": 1,
  "title": "Reproduction: A Graphop Analysis of Graph Neural Networks on Sparse Graphs: Generalization and Universal Approximation",
  "emoji": "🕸️",
  "space_id": "snaykey/repro-graphop-gnn",
  "paper": {
    "openreview_id": "tRsnpaRO0m"
  },
  "tags": [
    "icml2026-repro",
    "paper-tRsnpaRO0m"
  ],
  "updated_at": "2026-07-31T00:00:00Z",
  "root": {
    "slug": "index",
    "title": "Reproduction: A Graphop Analysis of Graph Neural Networks on Sparse Graphs: Generalization and Universal Approximation",
    "file": "pages/index.md",
    "children": [
      {
        "slug": "executive-summary",
        "title": "Executive summary",
        "file": "pages/executive-summary/page.md",
        "children": []
      },
      {
        "slug": "claim-1-def31-graphops",
        "title": "Graphops are introduced as self-adjoint, positivity-preserving operators over probability spaces, unifying dense graphons and sparse graphs as a single class of limit objects (Definition 3.1).",
        "file": "pages/claim-1-def31-graphops/page.md",
        "children": []
      },
      {
        "slug": "claim-2-def31-bofops",
        "title": "A restricted subclass called bounded-fiber operators (bofops) is defined via fiber measures nu_x satisfying ess sup_{x in Omega} nu_x(Omega) < infinity, capturing sparse connectivity patterns within the graphop framework (Definition 3.1).",
        "file": "pages/claim-2-def31-bofops/page.md",
        "children": []
      },
      {
        "slug": "claim-3-thm41-lipschitz",
        "title": "Theorem 4.1 establishes that message passing neural networks (MPNNs) are Lipschitz (Holder) continuous with respect to an action metric d_M on bofop-signals: ||H(A_1,f_1) - H(A_2,f_2)||_2 <= C'_{D,r} d_M((A_1,f_1),(A_2,f_2)) (Theorem 4.1).",
        "file": "pages/claim-3-thm41-lipschitz/page.md",
        "children": []
      },
      {
        "slug": "claim-4-cor53-compactness",
        "title": "Corollary 5.3 shows the space of bofop-DIDMs (degree-indexed distribution measures) is compact under the DIDM-mover's distance, forming a proper subset of the corresponding dense-graph structure (Corollary 5.3).",
        "file": "pages/claim-4-cor53-compactness/page.md",
        "children": []
      },
      {
        "slug": "claim-5-sec61-universal-approximation",
        "title": "Section 6.1 uses this compactness and continuity to prove a universal approximation theorem: any continuous function on bofop-DIDMs can be uniformly approximated by MPNNs directly on sparse graphs (Section 6.1).",
        "file": "pages/claim-5-sec61-universal-approximation/page.md",
        "children": []
      },
      {
        "slug": "claim-6-sec62-generalization",
        "title": "Section 6.2 derives generalization bounds showing generalization error vanishes as sample size grows, exploiting the uniform equicontinuity and compactness established in Theorem 4.1 and Corollary 5.3 (Section 6.2).",
        "file": "pages/claim-6-sec62-generalization/page.md",
        "children": []
      },
      {
        "slug": "conclusion",
        "title": "Conclusion",
        "file": "pages/conclusion/page.md",
        "children": []
      }
    ]
  }
}