{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:ABGZXP7I5OMTJCTGUKLKGCWMEN","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":"75544d15805a1ca264223f2baa67fb9358e98ff596c76687b3d16e2344f04bac","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2012-09-10T10:58:21Z","title_canon_sha256":"bfcaa05aad1f3ef1dc7ae3a7ea1d4db45b2d281665efcc2f07f457d141d94a71"},"schema_version":"1.0","source":{"id":"1209.1937","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.1937","created_at":"2026-05-18T03:45:54Z"},{"alias_kind":"arxiv_version","alias_value":"1209.1937v1","created_at":"2026-05-18T03:45:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.1937","created_at":"2026-05-18T03:45:54Z"},{"alias_kind":"pith_short_12","alias_value":"ABGZXP7I5OMT","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"ABGZXP7I5OMTJCTG","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"ABGZXP7I","created_at":"2026-05-18T12:26:58Z"}],"graph_snapshots":[{"event_id":"sha256:7a5ef59c81597d56b35968a78ad03f20a5447c9a4cb89a3f8c4a8ab4a123af76","target":"graph","created_at":"2026-05-18T03:45:54Z","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":"We study the agglomerative percolation (AP) models on the Bethe lattice and the triangular cactus to establish the exact mean-field theory for AP. Using the self-consistent simulation method, based on the exact self-consistent equation, we directly measure the order parameter $P_{\\infty}$ and average cluster size $S$. From the measured $P_{\\infty}$ and $S$ we obtain the critical exponents $\\beta_k$ and $\\gamma_k$ for $k=2$ and 3. Here $\\beta_k$ and $\\gamma_k$ are the critical exponents for $P_\\infty$ and $S$ when the growth of clusters spontaneously breaks the $Z_k$ symmetry of the $k$-partite","authors_text":"Huiseung Chae, Soon-Hyung Yook, Yup Kim","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2012-09-10T10:58:21Z","title":"Agglomerative percolation on the Bethe lattice and the triangular cactus"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.1937","kind":"arxiv","version":1},"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":"sha256:52fd2490fb38741f6d83eab7c2a74a3013e3142edee431a67d0ffc7e8fe8659e","target":"record","created_at":"2026-05-18T03:45:54Z","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":"75544d15805a1ca264223f2baa67fb9358e98ff596c76687b3d16e2344f04bac","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2012-09-10T10:58:21Z","title_canon_sha256":"bfcaa05aad1f3ef1dc7ae3a7ea1d4db45b2d281665efcc2f07f457d141d94a71"},"schema_version":"1.0","source":{"id":"1209.1937","kind":"arxiv","version":1}},"canonical_sha256":"004d9bbfe8eb99348a66a296a30acc2364eb037634bcdfa0b6614dd88482d172","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"004d9bbfe8eb99348a66a296a30acc2364eb037634bcdfa0b6614dd88482d172","first_computed_at":"2026-05-18T03:45:54.047425Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:45:54.047425Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GvJAxyKjVUWxZwsgifLSlO8DegZ76iMn8fE0aCNToTnE7nWBayRK/Q3R3NtqBGt+Mmvd541QhZ5o50amFlATAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:45:54.048011Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.1937","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:52fd2490fb38741f6d83eab7c2a74a3013e3142edee431a67d0ffc7e8fe8659e","sha256:7a5ef59c81597d56b35968a78ad03f20a5447c9a4cb89a3f8c4a8ab4a123af76"],"state_sha256":"159dbea8010496ad1694dd99098c5726e9803fdc6a186f75081c68954791e65d"}