{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:QHL4KG3K6BLO6UUARKEV63W7CA","short_pith_number":"pith:QHL4KG3K","canonical_record":{"source":{"id":"1201.4218","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2012-01-20T07:05:49Z","cross_cats_sorted":[],"title_canon_sha256":"65c1413484ac29aa0780f2fa3ff0e787b04b460176d5ee7202cc996227ae9423","abstract_canon_sha256":"906ce2c65663018d3b07079c6101ff8fb85f19a8d2d696c44be1e7be3f3654ba"},"schema_version":"1.0"},"canonical_sha256":"81d7c51b6af056ef52808a895f6edf101bd876a94844d742cc5b9f8a63cabcce","source":{"kind":"arxiv","id":"1201.4218","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.4218","created_at":"2026-05-18T01:58:46Z"},{"alias_kind":"arxiv_version","alias_value":"1201.4218v1","created_at":"2026-05-18T01:58:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.4218","created_at":"2026-05-18T01:58:46Z"},{"alias_kind":"pith_short_12","alias_value":"QHL4KG3K6BLO","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"QHL4KG3K6BLO6UUA","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"QHL4KG3K","created_at":"2026-05-18T12:27:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:QHL4KG3K6BLO6UUARKEV63W7CA","target":"record","payload":{"canonical_record":{"source":{"id":"1201.4218","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2012-01-20T07:05:49Z","cross_cats_sorted":[],"title_canon_sha256":"65c1413484ac29aa0780f2fa3ff0e787b04b460176d5ee7202cc996227ae9423","abstract_canon_sha256":"906ce2c65663018d3b07079c6101ff8fb85f19a8d2d696c44be1e7be3f3654ba"},"schema_version":"1.0"},"canonical_sha256":"81d7c51b6af056ef52808a895f6edf101bd876a94844d742cc5b9f8a63cabcce","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:58:46.391424Z","signature_b64":"gAQcipu+c/Ac5FElzM4ddN17keJubFX5d3+NkrvfveToeRAbs5d3nhfpxD9oYKjcFKfZcnQTYHUEUxLl9ap0CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"81d7c51b6af056ef52808a895f6edf101bd876a94844d742cc5b9f8a63cabcce","last_reissued_at":"2026-05-18T01:58:46.390665Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:58:46.390665Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1201.4218","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:58:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9H0oS8b/RSScJRuv9PRcoC4nScEuCYmY92BrQfn2HCTfQPbfBIyFmPL13NA9fOLyBD3XswkMezQ+tIdAwopSCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T18:21:07.174068Z"},"content_sha256":"e6fc2aba0587ee6da8c6d0dc83ca07fae416fd210943f58523826f8d848d7179","schema_version":"1.0","event_id":"sha256:e6fc2aba0587ee6da8c6d0dc83ca07fae416fd210943f58523826f8d848d7179"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:QHL4KG3K6BLO6UUARKEV63W7CA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Explosive percolations on the Bethe Lattice","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cond-mat.stat-mech","authors_text":"Huiseung Chae, Soon-Hyung Yook, Yup Kim","submitted_at":"2012-01-20T07:05:49Z","abstract_excerpt":"Based on the self-consistent equations of the order parameter $P_\\infty$ and the mean cluster size $S$, we develop a novel self-consistent simulation (SCS) method for arbitrary percolation on the Bethe lattice (infinite homogeneous Cayley tree). By applying SCS to the well-known percolation models, random bond percolation and bootstrap percolation, we obtain prototype functions for continuous and discontinuous phase transitions. By comparing the key functions obtained from SCSs for the Achlioptas processes (APs) with a product rule and a sum rule to the prototype functions, we show that the pe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.4218","kind":"arxiv","version":1},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:58:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HnAYaAO4Pb8a64+HxXBpBBidZzVnMAGHtF2Z7GYRwn5m7bNPkpeHOK5k1PQmm8c1KEkwJidXBoL+scL98bcDBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T18:21:07.174730Z"},"content_sha256":"2d399754eb9688be384c3e0e5bba4c245a1bd5f66ba3801b52f15665b8af393b","schema_version":"1.0","event_id":"sha256:2d399754eb9688be384c3e0e5bba4c245a1bd5f66ba3801b52f15665b8af393b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QHL4KG3K6BLO6UUARKEV63W7CA/bundle.json","state_url":"https://pith.science/pith/QHL4KG3K6BLO6UUARKEV63W7CA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QHL4KG3K6BLO6UUARKEV63W7CA/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-06T18:21:07Z","links":{"resolver":"https://pith.science/pith/QHL4KG3K6BLO6UUARKEV63W7CA","bundle":"https://pith.science/pith/QHL4KG3K6BLO6UUARKEV63W7CA/bundle.json","state":"https://pith.science/pith/QHL4KG3K6BLO6UUARKEV63W7CA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QHL4KG3K6BLO6UUARKEV63W7CA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:QHL4KG3K6BLO6UUARKEV63W7CA","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":"906ce2c65663018d3b07079c6101ff8fb85f19a8d2d696c44be1e7be3f3654ba","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2012-01-20T07:05:49Z","title_canon_sha256":"65c1413484ac29aa0780f2fa3ff0e787b04b460176d5ee7202cc996227ae9423"},"schema_version":"1.0","source":{"id":"1201.4218","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.4218","created_at":"2026-05-18T01:58:46Z"},{"alias_kind":"arxiv_version","alias_value":"1201.4218v1","created_at":"2026-05-18T01:58:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.4218","created_at":"2026-05-18T01:58:46Z"},{"alias_kind":"pith_short_12","alias_value":"QHL4KG3K6BLO","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"QHL4KG3K6BLO6UUA","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"QHL4KG3K","created_at":"2026-05-18T12:27:18Z"}],"graph_snapshots":[{"event_id":"sha256:2d399754eb9688be384c3e0e5bba4c245a1bd5f66ba3801b52f15665b8af393b","target":"graph","created_at":"2026-05-18T01:58:46Z","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":"Based on the self-consistent equations of the order parameter $P_\\infty$ and the mean cluster size $S$, we develop a novel self-consistent simulation (SCS) method for arbitrary percolation on the Bethe lattice (infinite homogeneous Cayley tree). By applying SCS to the well-known percolation models, random bond percolation and bootstrap percolation, we obtain prototype functions for continuous and discontinuous phase transitions. By comparing the key functions obtained from SCSs for the Achlioptas processes (APs) with a product rule and a sum rule to the prototype functions, we show that the pe","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":"2012-01-20T07:05:49Z","title":"Explosive percolations on the Bethe Lattice"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.4218","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:e6fc2aba0587ee6da8c6d0dc83ca07fae416fd210943f58523826f8d848d7179","target":"record","created_at":"2026-05-18T01:58:46Z","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":"906ce2c65663018d3b07079c6101ff8fb85f19a8d2d696c44be1e7be3f3654ba","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2012-01-20T07:05:49Z","title_canon_sha256":"65c1413484ac29aa0780f2fa3ff0e787b04b460176d5ee7202cc996227ae9423"},"schema_version":"1.0","source":{"id":"1201.4218","kind":"arxiv","version":1}},"canonical_sha256":"81d7c51b6af056ef52808a895f6edf101bd876a94844d742cc5b9f8a63cabcce","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"81d7c51b6af056ef52808a895f6edf101bd876a94844d742cc5b9f8a63cabcce","first_computed_at":"2026-05-18T01:58:46.390665Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:58:46.390665Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gAQcipu+c/Ac5FElzM4ddN17keJubFX5d3+NkrvfveToeRAbs5d3nhfpxD9oYKjcFKfZcnQTYHUEUxLl9ap0CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:58:46.391424Z","signed_message":"canonical_sha256_bytes"},"source_id":"1201.4218","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e6fc2aba0587ee6da8c6d0dc83ca07fae416fd210943f58523826f8d848d7179","sha256:2d399754eb9688be384c3e0e5bba4c245a1bd5f66ba3801b52f15665b8af393b"],"state_sha256":"6bf93086c2003272288fa6c19bc84da4fa8238182d4dfa4adedddd811ef6c945"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YFGk5Sa0EKNro3zJMcJXLEx7HqaYfHmqKbAq+Ff1vRLGTafXpGJqWx7c8qJPzCJ9+omatY7m3A98nZEa+E20DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T18:21:07.178379Z","bundle_sha256":"7e54caed41cb2934552cd95e2483b70aaf7eb3cbc7ed269efa2f37c2b8a55468"}}