{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:V3SJEDH4XOKGFEVB7HJTNOPPGZ","short_pith_number":"pith:V3SJEDH4","schema_version":"1.0","canonical_sha256":"aee4920cfcbb946292a1f9d336b9ef367d75d85fe9db89546a146a094c0dc1b8","source":{"kind":"arxiv","id":"1405.0599","version":3},"attestation_state":"computed","paper":{"title":"Multipodal Structure and Phase Transitions in Large Constrained Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.SI","math-ph","math.MP","math.PR"],"primary_cat":"math.CO","authors_text":"Charles Radin, Kui Ren, Lorenzo Sadun, Richard Kenyon","submitted_at":"2014-05-03T15:06:49Z","abstract_excerpt":"We study the asymptotics of large, simple, labeled graphs constrained by the densities of edges and of $k$-star subgraphs, $k\\ge 2$ fixed. We prove that under such constraints graphs are \"multipodal\": asymptotically in the number of vertices there is a partition of the vertices into $M < \\infty$ subsets $V_1, V_2, \\ldots, V_M$, and a set of well-defined probabilities $g_{ij}$ of an edge between any $v_i \\in V_i$ and $v_j \\in V_j$. For $2\\le k\\le 30$ we determine the phase space: the combinations of edge and $k$-star densities achievable asymptotically. For these models there are special points"},"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":"1405.0599","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-05-03T15:06:49Z","cross_cats_sorted":["cs.SI","math-ph","math.MP","math.PR"],"title_canon_sha256":"2177b674546cad584994278af9e033f1e5aa029b3e1f6b389e677b045977d450","abstract_canon_sha256":"2763afb9d08fc8c9a91e69e77e4d68b42b65ee08e0e0b68c057fc97143852541"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:48:43.454064Z","signature_b64":"kk7XsTEqfV/MGkea5Io5f/jEch2Cw3y+78ztft5QRxi+JVt5fX52zVe/XFwyyH2ga//mDUJH/DwzCi87Grk+Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aee4920cfcbb946292a1f9d336b9ef367d75d85fe9db89546a146a094c0dc1b8","last_reissued_at":"2026-05-18T00:48:43.453632Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:48:43.453632Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multipodal Structure and Phase Transitions in Large Constrained Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.SI","math-ph","math.MP","math.PR"],"primary_cat":"math.CO","authors_text":"Charles Radin, Kui Ren, Lorenzo Sadun, Richard Kenyon","submitted_at":"2014-05-03T15:06:49Z","abstract_excerpt":"We study the asymptotics of large, simple, labeled graphs constrained by the densities of edges and of $k$-star subgraphs, $k\\ge 2$ fixed. We prove that under such constraints graphs are \"multipodal\": asymptotically in the number of vertices there is a partition of the vertices into $M < \\infty$ subsets $V_1, V_2, \\ldots, V_M$, and a set of well-defined probabilities $g_{ij}$ of an edge between any $v_i \\in V_i$ and $v_j \\in V_j$. For $2\\le k\\le 30$ we determine the phase space: the combinations of edge and $k$-star densities achievable asymptotically. For these models there are special points"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.0599","kind":"arxiv","version":3},"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":"1405.0599","created_at":"2026-05-18T00:48:43.453697+00:00"},{"alias_kind":"arxiv_version","alias_value":"1405.0599v3","created_at":"2026-05-18T00:48:43.453697+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.0599","created_at":"2026-05-18T00:48:43.453697+00:00"},{"alias_kind":"pith_short_12","alias_value":"V3SJEDH4XOKG","created_at":"2026-05-18T12:28:52.271510+00:00"},{"alias_kind":"pith_short_16","alias_value":"V3SJEDH4XOKGFEVB","created_at":"2026-05-18T12:28:52.271510+00:00"},{"alias_kind":"pith_short_8","alias_value":"V3SJEDH4","created_at":"2026-05-18T12:28:52.271510+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/V3SJEDH4XOKGFEVB7HJTNOPPGZ","json":"https://pith.science/pith/V3SJEDH4XOKGFEVB7HJTNOPPGZ.json","graph_json":"https://pith.science/api/pith-number/V3SJEDH4XOKGFEVB7HJTNOPPGZ/graph.json","events_json":"https://pith.science/api/pith-number/V3SJEDH4XOKGFEVB7HJTNOPPGZ/events.json","paper":"https://pith.science/paper/V3SJEDH4"},"agent_actions":{"view_html":"https://pith.science/pith/V3SJEDH4XOKGFEVB7HJTNOPPGZ","download_json":"https://pith.science/pith/V3SJEDH4XOKGFEVB7HJTNOPPGZ.json","view_paper":"https://pith.science/paper/V3SJEDH4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1405.0599&json=true","fetch_graph":"https://pith.science/api/pith-number/V3SJEDH4XOKGFEVB7HJTNOPPGZ/graph.json","fetch_events":"https://pith.science/api/pith-number/V3SJEDH4XOKGFEVB7HJTNOPPGZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/V3SJEDH4XOKGFEVB7HJTNOPPGZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/V3SJEDH4XOKGFEVB7HJTNOPPGZ/action/storage_attestation","attest_author":"https://pith.science/pith/V3SJEDH4XOKGFEVB7HJTNOPPGZ/action/author_attestation","sign_citation":"https://pith.science/pith/V3SJEDH4XOKGFEVB7HJTNOPPGZ/action/citation_signature","submit_replication":"https://pith.science/pith/V3SJEDH4XOKGFEVB7HJTNOPPGZ/action/replication_record"}},"created_at":"2026-05-18T00:48:43.453697+00:00","updated_at":"2026-05-18T00:48:43.453697+00:00"}