{"paper":{"title":"Star-collision in random hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Nontrivial units disappear with high probability in random hypergraphs under specific regimes.","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Kartick Adhikari, Samiron Parui","submitted_at":"2026-05-16T07:41:14Z","abstract_excerpt":"We study star-based symmetries in uniform hypergraphs and their consequences for matrices whose entries depend only on vertex stars. Such matrices admit a deterministic decomposition into a global component and a local component supported on equivalence classes of vertices with identical stars, known as units. While nontrivial units may exist at finite size in hypergraphs of uniformity greater than two, their persistence in random settings has remained unclear.\n  We analyze star collisions in random $k$-uniform hypergraphs and show that, in some particular regimes, nontrivial units disappear w"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"nontrivial units disappear with high probability as the number of vertices grows. As a consequence, star-dependent matrices exhibit asymptotically trivial local structure, and their spectral behavior, invariant subspaces, and associated linear dynamics are governed by a reduced quotient object obtained by contracting vertex stars.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The disappearance holds only in some particular regimes of the random k-uniform hypergraph model (specific ranges of uniformity k and edge probability).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"In random k-uniform hypergraphs, star collisions cause nontrivial units to disappear with high probability, yielding asymptotically trivial local structure for star-dependent matrices governed by a quotient object.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Nontrivial units disappear with high probability in random hypergraphs under specific regimes.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"033d28cfcbba8fcc259f0d09b20b5be0d3315903262c82b808ec44b38b033beb"},"source":{"id":"2605.16856","kind":"arxiv","version":1},"verdict":{"id":"3b2a7f43-97bb-4253-bc42-d4b07f3581f4","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T20:42:05.016709Z","strongest_claim":"nontrivial units disappear with high probability as the number of vertices grows. As a consequence, star-dependent matrices exhibit asymptotically trivial local structure, and their spectral behavior, invariant subspaces, and associated linear dynamics are governed by a reduced quotient object obtained by contracting vertex stars.","one_line_summary":"In random k-uniform hypergraphs, star collisions cause nontrivial units to disappear with high probability, yielding asymptotically trivial local structure for star-dependent matrices governed by a quotient object.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The disappearance holds only in some particular regimes of the random k-uniform hypergraph model (specific ranges of uniformity k and edge probability).","pith_extraction_headline":"Nontrivial units disappear with high probability in random hypergraphs under specific regimes."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16856/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T21:01:19.239525Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T20:51:12.330858Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T18:41:56.309417Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T18:33:26.384037Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"bf497eb0f2c53381e20619d6b8b1da49b8eed236566680504169eafeb97ffc13"},"references":{"count":38,"sample":[{"doi":"","year":2025,"title":"K. Adhikari and A. Khatun , On the diameter of random uniform hypergraphs in dense regime , arXiv preprint arXiv:2512.04544, (2025)","work_id":"007f5ab1-a3c3-4c9c-8c54-7352d82c2589","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"K. Adhikari and S. Parui , Spectrum and local weak convergence of sparse random uniform hypergraphs , arXiv preprint arXiv:2509.05102, (2025)","work_id":"1b1ef2f3-7d55-4f84-9bc3-e7428052474d","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2004,"title":"D. Aldous and J. M. Steele , The objective method: probabilistic combinatorial optimization and local weak convergence , in Probability on discrete structures, Springer, 2004, pp. 1--72","work_id":"87eac7bf-8c47-4a92-9846-6efa384f746b","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"Banerjee , On the spectrum of hypergraphs , Linear Algebra and its Applications, 614 (2021), pp","work_id":"76b0421b-a95d-4300-aa3e-4c303b7e3b53","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"title":"A. Banerjee, A. Char, and B. Mondal , Spectra of general hypergraphs , Linear Algebra Appl., 518 (2017), pp. 14--30","work_id":"042eb44c-af7d-43e1-8301-1dd3287e2639","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":38,"snapshot_sha256":"f5503b5ee81add9d18e105cfb9cab8cc8863a3c37d79095c183910ef496c7d5b","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"89ccaefb9a4a351024110e19a39227e62cb443ee633b9434c01c7124b9a8d67a"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}