{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:NIRMHK76325FUR7PWY4GCWQZDV","short_pith_number":"pith:NIRMHK76","canonical_record":{"source":{"id":"1806.10975","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-28T13:57:41Z","cross_cats_sorted":[],"title_canon_sha256":"b25163912580a6d0159a3ba9934694f453ac77d9fafabd599f0385fb636f6f4a","abstract_canon_sha256":"e536f8846aa7a25c157950389b7ed36f5994ca53fd9571e64d8a40e2b963cea1"},"schema_version":"1.0"},"canonical_sha256":"6a22c3abfedeba5a47efb638615a191d51cbb581369053c4e9fd1d7bee1b8784","source":{"kind":"arxiv","id":"1806.10975","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.10975","created_at":"2026-05-18T00:12:07Z"},{"alias_kind":"arxiv_version","alias_value":"1806.10975v1","created_at":"2026-05-18T00:12:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.10975","created_at":"2026-05-18T00:12:07Z"},{"alias_kind":"pith_short_12","alias_value":"NIRMHK76325F","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_16","alias_value":"NIRMHK76325FUR7P","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_8","alias_value":"NIRMHK76","created_at":"2026-05-18T12:32:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:NIRMHK76325FUR7PWY4GCWQZDV","target":"record","payload":{"canonical_record":{"source":{"id":"1806.10975","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-28T13:57:41Z","cross_cats_sorted":[],"title_canon_sha256":"b25163912580a6d0159a3ba9934694f453ac77d9fafabd599f0385fb636f6f4a","abstract_canon_sha256":"e536f8846aa7a25c157950389b7ed36f5994ca53fd9571e64d8a40e2b963cea1"},"schema_version":"1.0"},"canonical_sha256":"6a22c3abfedeba5a47efb638615a191d51cbb581369053c4e9fd1d7bee1b8784","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:07.371350Z","signature_b64":"sP2HarBJhMgWg5bluykulVYvLPDgfhi4AW1MZJb64xPqxyvpDmCXsifAdgEG/9rn+dDkalIx1gAa9SxffAOFBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6a22c3abfedeba5a47efb638615a191d51cbb581369053c4e9fd1d7bee1b8784","last_reissued_at":"2026-05-18T00:12:07.370636Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:07.370636Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1806.10975","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":"2026-05-18T00:12:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WiaCBrhId8FkqVzitsFve/dGeUbCOC8kd0PgaJRf0LgNc0OTj7xMjwiFveHisFzeKoLraj7DIV/O52o2oqxpCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T20:31:27.939817Z"},"content_sha256":"b88e059fc34926a6fc65d06134d82b4c8507f719e2d2a1188ed8a340865a0bf8","schema_version":"1.0","event_id":"sha256:b88e059fc34926a6fc65d06134d82b4c8507f719e2d2a1188ed8a340865a0bf8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:NIRMHK76325FUR7PWY4GCWQZDV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A variant of the Erdos-Renyi random graph process","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adam Logan, Mike Molloy, Pawel Pralat","submitted_at":"2018-06-28T13:57:41Z","abstract_excerpt":"We consider a natural variant of the Erd\\H{o}s-R\\'enyi random graph process in which $k$ vertices are special and are never put into the same connected component. The model is natural and interesting on its own, but is actually inspired by the combinatorial data fusion problem that itself is connected to a number of important problems in graph theory. We will show that a phase transition occurs when the number of special vertices is roughly $n^{1/3}$, where $n$ is the number of vertices."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.10975","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":"2026-05-18T00:12:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"W3f1vqJuSyKMZIc1xfKc8rH3xpp3A61C9HOJtqOAYJT0HLUCGEV0io+PGHBJDOjAazDJ38wHdc9Dq+XnofaODw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T20:31:27.940145Z"},"content_sha256":"901733fa691de55e29f6e2258fb1bfefb6d402d3ca1efb0b493e76922b1b0c58","schema_version":"1.0","event_id":"sha256:901733fa691de55e29f6e2258fb1bfefb6d402d3ca1efb0b493e76922b1b0c58"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NIRMHK76325FUR7PWY4GCWQZDV/bundle.json","state_url":"https://pith.science/pith/NIRMHK76325FUR7PWY4GCWQZDV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NIRMHK76325FUR7PWY4GCWQZDV/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":"2026-06-21T20:31:27Z","links":{"resolver":"https://pith.science/pith/NIRMHK76325FUR7PWY4GCWQZDV","bundle":"https://pith.science/pith/NIRMHK76325FUR7PWY4GCWQZDV/bundle.json","state":"https://pith.science/pith/NIRMHK76325FUR7PWY4GCWQZDV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NIRMHK76325FUR7PWY4GCWQZDV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:NIRMHK76325FUR7PWY4GCWQZDV","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":"e536f8846aa7a25c157950389b7ed36f5994ca53fd9571e64d8a40e2b963cea1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-28T13:57:41Z","title_canon_sha256":"b25163912580a6d0159a3ba9934694f453ac77d9fafabd599f0385fb636f6f4a"},"schema_version":"1.0","source":{"id":"1806.10975","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.10975","created_at":"2026-05-18T00:12:07Z"},{"alias_kind":"arxiv_version","alias_value":"1806.10975v1","created_at":"2026-05-18T00:12:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.10975","created_at":"2026-05-18T00:12:07Z"},{"alias_kind":"pith_short_12","alias_value":"NIRMHK76325F","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_16","alias_value":"NIRMHK76325FUR7P","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_8","alias_value":"NIRMHK76","created_at":"2026-05-18T12:32:40Z"}],"graph_snapshots":[{"event_id":"sha256:901733fa691de55e29f6e2258fb1bfefb6d402d3ca1efb0b493e76922b1b0c58","target":"graph","created_at":"2026-05-18T00:12:07Z","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 consider a natural variant of the Erd\\H{o}s-R\\'enyi random graph process in which $k$ vertices are special and are never put into the same connected component. The model is natural and interesting on its own, but is actually inspired by the combinatorial data fusion problem that itself is connected to a number of important problems in graph theory. We will show that a phase transition occurs when the number of special vertices is roughly $n^{1/3}$, where $n$ is the number of vertices.","authors_text":"Adam Logan, Mike Molloy, Pawel Pralat","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-28T13:57:41Z","title":"A variant of the Erdos-Renyi random graph process"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.10975","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:b88e059fc34926a6fc65d06134d82b4c8507f719e2d2a1188ed8a340865a0bf8","target":"record","created_at":"2026-05-18T00:12:07Z","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":"e536f8846aa7a25c157950389b7ed36f5994ca53fd9571e64d8a40e2b963cea1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-28T13:57:41Z","title_canon_sha256":"b25163912580a6d0159a3ba9934694f453ac77d9fafabd599f0385fb636f6f4a"},"schema_version":"1.0","source":{"id":"1806.10975","kind":"arxiv","version":1}},"canonical_sha256":"6a22c3abfedeba5a47efb638615a191d51cbb581369053c4e9fd1d7bee1b8784","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6a22c3abfedeba5a47efb638615a191d51cbb581369053c4e9fd1d7bee1b8784","first_computed_at":"2026-05-18T00:12:07.370636Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:12:07.370636Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sP2HarBJhMgWg5bluykulVYvLPDgfhi4AW1MZJb64xPqxyvpDmCXsifAdgEG/9rn+dDkalIx1gAa9SxffAOFBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:12:07.371350Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.10975","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b88e059fc34926a6fc65d06134d82b4c8507f719e2d2a1188ed8a340865a0bf8","sha256:901733fa691de55e29f6e2258fb1bfefb6d402d3ca1efb0b493e76922b1b0c58"],"state_sha256":"f76c9cb44f5f633a9ab44ede42b4d7bfc5b3c0e2574777143bb058fb6cc5dca2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sMjoflwfIL01Dn3eq/7W5BpC6jDgtVFqHhfBagofZkwPKktRLkPXt0ymVeth/rG7ytAESIAzfBoblGE3RFPqDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T20:31:27.942035Z","bundle_sha256":"05d2acef1ffa07fd34e47196d8ff160903fc3e741d74e3764a64b36a3fb0c9f4"}}