{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:UE7YMQJEYA6OILMCO7V3LL4TA6","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"e076a2f1b701d04ce2a7c1f4a0d55f6d9a44df84646ea3e2afb6f3ce176d0099","cross_cats_sorted":["cond-mat.stat-mech","cs.SI"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"physics.soc-ph","submitted_at":"2016-07-28T20:44:00Z","title_canon_sha256":"20de6cf9eccafae3d28621b27f9273e6fabdcb9e78ead115216d9722a214c0a1"},"schema_version":"1.0","source":{"id":"1607.08637","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.08637","created_at":"2026-05-18T00:52:09Z"},{"alias_kind":"arxiv_version","alias_value":"1607.08637v2","created_at":"2026-05-18T00:52:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.08637","created_at":"2026-05-18T00:52:09Z"},{"alias_kind":"pith_short_12","alias_value":"UE7YMQJEYA6O","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"UE7YMQJEYA6OILMC","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"UE7YMQJE","created_at":"2026-05-18T12:30:46Z"}],"graph_snapshots":[{"event_id":"sha256:f7a1b3f70f38421cfd7962a19e376d78575819e2db87a2be362f7c8d37686b88","target":"graph","created_at":"2026-05-18T00:52:09Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"The giant $k$-core --- maximal connected subgraph of a network where each node has at least $k$ neighbors --- is important in the study of phase transitions and in applications of network theory. Unlike Erd\\H{o}s-R\\'enyi graphs and other random networks where $k$-cores emerge discontinuously for $k\\ge 3$, we show that transitive linking (or triadic closure) leads to 3-cores emerging through single or double phase transitions of both discontinuous and continuous nature. We also develop a $k$-core calculation that includes clustering and provides insights into how high-level connectivity emerges","authors_text":"Laurent H\\'ebert-Dufresne, Munik Shrestha, Uttam Bhat","cross_cats":["cond-mat.stat-mech","cs.SI"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"physics.soc-ph","submitted_at":"2016-07-28T20:44:00Z","title":"Exotic phase transitions of k-cores in clustered networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.08637","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:254b403b20b63dc03e819c8b037d802787b0e405587e1c25380b80009438d517","target":"record","created_at":"2026-05-18T00:52:09Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"e076a2f1b701d04ce2a7c1f4a0d55f6d9a44df84646ea3e2afb6f3ce176d0099","cross_cats_sorted":["cond-mat.stat-mech","cs.SI"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"physics.soc-ph","submitted_at":"2016-07-28T20:44:00Z","title_canon_sha256":"20de6cf9eccafae3d28621b27f9273e6fabdcb9e78ead115216d9722a214c0a1"},"schema_version":"1.0","source":{"id":"1607.08637","kind":"arxiv","version":2}},"canonical_sha256":"a13f864124c03ce42d8277ebb5af930782b403b068ba7379565ecbf506a6c6e8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a13f864124c03ce42d8277ebb5af930782b403b068ba7379565ecbf506a6c6e8","first_computed_at":"2026-05-18T00:52:09.385595Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:52:09.385595Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"26TkYLe2hcSNeAGMCu2v18GG5Do1erraoKH16kb3dpWMb2d99GIM//OhCHw3CixrahC/607UCoV5rHsepZcjAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:52:09.386273Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.08637","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:254b403b20b63dc03e819c8b037d802787b0e405587e1c25380b80009438d517","sha256:f7a1b3f70f38421cfd7962a19e376d78575819e2db87a2be362f7c8d37686b88"],"state_sha256":"57f501c1b54edb940e6f05dd31ed6f12d3f4504e5a17b3735d46696bc22ae1ee"}