{
  "schema_version": 1,
  "problem_number": "OWR-15962-003",
  "title": "Seed Aggregation for Competing First-Passage Percolation",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We give an exact reduction from competing first-passage percolation with initially infected hubs to the same process with many seeds on a regular configuration model. Merging vertices already occupied by the same type preserves the entire infection process outside those vertices. Combining this observation with the regular-graph theorem of Antunovic, Dekel, Mossel and Peres yields a sharp seed-count criterion within an almost regular family. If the slow-to-fast rate ratio is r in (0,1) and there are q_N=o(N) slow hubs, a construction with convergent finite second degree moment in which the slow type captures asymptotically all vertices exists exactly when q_N/N^(1-2r) tends to infinity. One diverging-degree hub suffices precisely when r>1/2; at r=1/2, arbitrarily slowly diverging hub counts suffice. The high-probability conclusions also hold for the corresponding uniform simple graphs. The growth estimates are imported from prior work. The result concerns deterministic degree sequences with planted hubs and does not settle the question for a prescribed fixed iid degree law. This is an unrefereed, AI-assisted preprint without an absolute priority claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "keywords": [
    "OWR-15962-003",
    "competing first-passage percolation",
    "configuration model",
    "seed aggregation",
    "finite degree variance"
  ],
  "manuscript_version_date": "2026-10-25",
  "publication_date": "2026-10-25",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-25",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-15962-003/",
  "pdf_url": "https://eulersolve.org/papers/owr-15962-003/paper.pdf?v=9703a9948489",
  "doi": "10.5281/zenodo.23628465",
  "zenodo_record_url": "https://zenodo.org/records/23628465",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For an almost regular configuration-model family obtained by merging k_N slow seeds into q_N hubs, with fixed degree d>=3 and rate ratio r in (0,1), a construction with convergent finite second degree moment and slow-type final density tending to one exists exactly when q_N/N^(1-2r) tends to infinity, assuming q_N=o(N). This includes a sharp one-hub threshold r>1/2 and an explicit construction for every positive rate ratio. The high-probability conclusions transfer to uniform simple graphs. This deterministic planted-hub criterion does not settle the fixed-iid degree-law question or arbitrary finite-variance networks. All growth estimates are credited to Antunovic, Dekel, Mossel and Peres; Janson supplies the simplicity input. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "9703a99484892c1b2b196dc1817ae629899242b115efb9b4830e7d3e56b850af"
    },
    "source.zip": {
      "sha256": "b13cdb30eb75654058d419f874d9c7515f212d5179ec37e09771f38593269acd"
    },
    "verification_report.md": {
      "sha256": "03b71dba6115e9feada72f3f34b594d6fdc81de4aa640bc01c6802aab8bdd8f4"
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}