{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:HCUKYH7RGR64OR6JPCHBSSDTRY","short_pith_number":"pith:HCUKYH7R","schema_version":"1.0","canonical_sha256":"38a8ac1ff1347dc747c9788e1948738e28a27b354c393d6f7e76a7bfd3b08917","source":{"kind":"arxiv","id":"1409.1174","version":2},"attestation_state":"computed","paper":{"title":"High-order Phase Transition in Random Hypergrpahs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Linyuan Lu, Xing Peng","submitted_at":"2014-09-03T17:23:13Z","abstract_excerpt":"In this paper, we study the high-order phase transition in random $r$-uniform hypergraphs. For a positive integer $n$ and a real $p\\in [0,1]$, let $H:=H^r(n,p)$ be the random $r$-uniform hypergraph with vertex set $[n]$, where each $r$-set is selected as an edge with probability $p$ independently randomly. For $1\\leq s \\leq r-1$ and two $s$-sets $S$ and $S'$, we say $S$ is connected to $S'$ if there is a sequence of alternating $s$-sets and edges $S_0,F_1,S_1,F_2, \\ldots, F_k, S_k$ such that $S_0,S_1,\\ldots, S_k$ are $s$-sets, $S_0=S$, $S_k=S'$, $F_1,F_2,\\ldots, F_k$ are edges of $H$, and $S_{"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1409.1174","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-09-03T17:23:13Z","cross_cats_sorted":[],"title_canon_sha256":"4948029de28326731082243aa7b5e34447145a50a6f58dc207fef13f3aa98139","abstract_canon_sha256":"90a39abd23b47afaf3c8e53277371d00b66367190ab9e692aa5d82b687f235d6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:06.014328Z","signature_b64":"HYZMUyyj6xTcF7vR7+PSbUT+jqRn1MnkAr1XR3EmOeg+wPF8bMPxAqxuZc5lDgDJDgwasjRUVyYZZrL7JEHzDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"38a8ac1ff1347dc747c9788e1948738e28a27b354c393d6f7e76a7bfd3b08917","last_reissued_at":"2026-05-18T00:09:06.013770Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:06.013770Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"High-order Phase Transition in Random Hypergrpahs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Linyuan Lu, Xing Peng","submitted_at":"2014-09-03T17:23:13Z","abstract_excerpt":"In this paper, we study the high-order phase transition in random $r$-uniform hypergraphs. For a positive integer $n$ and a real $p\\in [0,1]$, let $H:=H^r(n,p)$ be the random $r$-uniform hypergraph with vertex set $[n]$, where each $r$-set is selected as an edge with probability $p$ independently randomly. For $1\\leq s \\leq r-1$ and two $s$-sets $S$ and $S'$, we say $S$ is connected to $S'$ if there is a sequence of alternating $s$-sets and edges $S_0,F_1,S_1,F_2, \\ldots, F_k, S_k$ such that $S_0,S_1,\\ldots, S_k$ are $s$-sets, $S_0=S$, $S_k=S'$, $F_1,F_2,\\ldots, F_k$ are edges of $H$, and $S_{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.1174","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1409.1174","created_at":"2026-05-18T00:09:06.013843+00:00"},{"alias_kind":"arxiv_version","alias_value":"1409.1174v2","created_at":"2026-05-18T00:09:06.013843+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.1174","created_at":"2026-05-18T00:09:06.013843+00:00"},{"alias_kind":"pith_short_12","alias_value":"HCUKYH7RGR64","created_at":"2026-05-18T12:28:30.664211+00:00"},{"alias_kind":"pith_short_16","alias_value":"HCUKYH7RGR64OR6J","created_at":"2026-05-18T12:28:30.664211+00:00"},{"alias_kind":"pith_short_8","alias_value":"HCUKYH7R","created_at":"2026-05-18T12:28:30.664211+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/HCUKYH7RGR64OR6JPCHBSSDTRY","json":"https://pith.science/pith/HCUKYH7RGR64OR6JPCHBSSDTRY.json","graph_json":"https://pith.science/api/pith-number/HCUKYH7RGR64OR6JPCHBSSDTRY/graph.json","events_json":"https://pith.science/api/pith-number/HCUKYH7RGR64OR6JPCHBSSDTRY/events.json","paper":"https://pith.science/paper/HCUKYH7R"},"agent_actions":{"view_html":"https://pith.science/pith/HCUKYH7RGR64OR6JPCHBSSDTRY","download_json":"https://pith.science/pith/HCUKYH7RGR64OR6JPCHBSSDTRY.json","view_paper":"https://pith.science/paper/HCUKYH7R","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1409.1174&json=true","fetch_graph":"https://pith.science/api/pith-number/HCUKYH7RGR64OR6JPCHBSSDTRY/graph.json","fetch_events":"https://pith.science/api/pith-number/HCUKYH7RGR64OR6JPCHBSSDTRY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HCUKYH7RGR64OR6JPCHBSSDTRY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HCUKYH7RGR64OR6JPCHBSSDTRY/action/storage_attestation","attest_author":"https://pith.science/pith/HCUKYH7RGR64OR6JPCHBSSDTRY/action/author_attestation","sign_citation":"https://pith.science/pith/HCUKYH7RGR64OR6JPCHBSSDTRY/action/citation_signature","submit_replication":"https://pith.science/pith/HCUKYH7RGR64OR6JPCHBSSDTRY/action/replication_record"}},"created_at":"2026-05-18T00:09:06.013843+00:00","updated_at":"2026-05-18T00:09:06.013843+00:00"}