{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:U4GE2PKL756TKSWHLZ2RVCXRZM","short_pith_number":"pith:U4GE2PKL","canonical_record":{"source":{"id":"2406.06380","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-06-10T15:40:25Z","cross_cats_sorted":[],"title_canon_sha256":"142492a4eb73cf6766d8cc6229839b60d5bae9db5241132ada08592ed92e84ff","abstract_canon_sha256":"0d4f1fb553cf49395fb9f5324d10fa3db5a27ce9f3ea44d18c86d4ee8df4aa5b"},"schema_version":"1.0"},"canonical_sha256":"a70c4d3d4bff7d354ac75e751a8af1cb12cdde93a9a71413a17392b4a4186400","source":{"kind":"arxiv","id":"2406.06380","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2406.06380","created_at":"2026-05-20T00:02:46Z"},{"alias_kind":"arxiv_version","alias_value":"2406.06380v2","created_at":"2026-05-20T00:02:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2406.06380","created_at":"2026-05-20T00:02:46Z"},{"alias_kind":"pith_short_12","alias_value":"U4GE2PKL756T","created_at":"2026-05-20T00:02:46Z"},{"alias_kind":"pith_short_16","alias_value":"U4GE2PKL756TKSWH","created_at":"2026-05-20T00:02:46Z"},{"alias_kind":"pith_short_8","alias_value":"U4GE2PKL","created_at":"2026-05-20T00:02:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:U4GE2PKL756TKSWHLZ2RVCXRZM","target":"record","payload":{"canonical_record":{"source":{"id":"2406.06380","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-06-10T15:40:25Z","cross_cats_sorted":[],"title_canon_sha256":"142492a4eb73cf6766d8cc6229839b60d5bae9db5241132ada08592ed92e84ff","abstract_canon_sha256":"0d4f1fb553cf49395fb9f5324d10fa3db5a27ce9f3ea44d18c86d4ee8df4aa5b"},"schema_version":"1.0"},"canonical_sha256":"a70c4d3d4bff7d354ac75e751a8af1cb12cdde93a9a71413a17392b4a4186400","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:02:46.143897Z","signature_b64":"QtQG2Qc2YqaB8m0k/xilQ2y1zirh2qN5bgL6SdQH5krb4K2HuSkXraOlJVGXj81rm4b0C9PMFCNVVXbaZMmTDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a70c4d3d4bff7d354ac75e751a8af1cb12cdde93a9a71413a17392b4a4186400","last_reissued_at":"2026-05-20T00:02:46.143262Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:02:46.143262Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2406.06380","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-20T00:02:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xFZZgPqX4ydfoIhOwelOBzWVN4fq1IDazuZNXaMkOJlFi46IS6cCufcFWsmPUAvoSP3Ytjz6L7a9KawJwx+tAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T22:04:56.422491Z"},"content_sha256":"ddec68c0a8052089ecb4b2d8864e3a84192bc7075bfbd2af869b896c667e7a95","schema_version":"1.0","event_id":"sha256:ddec68c0a8052089ecb4b2d8864e3a84192bc7075bfbd2af869b896c667e7a95"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:U4GE2PKL756TKSWHLZ2RVCXRZM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The number of connected components in sub-critical random graph processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Josu\\'e Corujo","submitted_at":"2024-06-10T15:40:25Z","abstract_excerpt":"We present a detailed study of the evolution of the number of connected components in sub-critical multiplicative random graph processes. We consider a model where edges appear independently after an exponential time at rate equal to the product of the sizes of the vertices. We provide an explicit expression for the fluid limit of the number of connected components normalized by its initial value, when the time is smaller than the inverse of the sum of the square of the initial vertex sizes. We also identify the diffusion limit of the rescaled fluctuations around the fluid limit. This is appli"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2406.06380","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2406.06380/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-05-20T00:02:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cF7IGEHw08ZHe/hRuaA4G35ZarQNyc4UkcWqdvL5Q+UTNmcvF9/EN1hHNAstCKh3jzkojTDjwGaHBBrgOfCrCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T22:04:56.423051Z"},"content_sha256":"b754620ec13fa2201264e675e5b49c41b3eb6055fad9ec870248a4a65c63e463","schema_version":"1.0","event_id":"sha256:b754620ec13fa2201264e675e5b49c41b3eb6055fad9ec870248a4a65c63e463"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/U4GE2PKL756TKSWHLZ2RVCXRZM/bundle.json","state_url":"https://pith.science/pith/U4GE2PKL756TKSWHLZ2RVCXRZM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/U4GE2PKL756TKSWHLZ2RVCXRZM/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":"2026-08-07T22:04:56Z","links":{"resolver":"https://pith.science/pith/U4GE2PKL756TKSWHLZ2RVCXRZM","bundle":"https://pith.science/pith/U4GE2PKL756TKSWHLZ2RVCXRZM/bundle.json","state":"https://pith.science/pith/U4GE2PKL756TKSWHLZ2RVCXRZM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/U4GE2PKL756TKSWHLZ2RVCXRZM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:U4GE2PKL756TKSWHLZ2RVCXRZM","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":"0d4f1fb553cf49395fb9f5324d10fa3db5a27ce9f3ea44d18c86d4ee8df4aa5b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-06-10T15:40:25Z","title_canon_sha256":"142492a4eb73cf6766d8cc6229839b60d5bae9db5241132ada08592ed92e84ff"},"schema_version":"1.0","source":{"id":"2406.06380","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2406.06380","created_at":"2026-05-20T00:02:46Z"},{"alias_kind":"arxiv_version","alias_value":"2406.06380v2","created_at":"2026-05-20T00:02:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2406.06380","created_at":"2026-05-20T00:02:46Z"},{"alias_kind":"pith_short_12","alias_value":"U4GE2PKL756T","created_at":"2026-05-20T00:02:46Z"},{"alias_kind":"pith_short_16","alias_value":"U4GE2PKL756TKSWH","created_at":"2026-05-20T00:02:46Z"},{"alias_kind":"pith_short_8","alias_value":"U4GE2PKL","created_at":"2026-05-20T00:02:46Z"}],"graph_snapshots":[{"event_id":"sha256:b754620ec13fa2201264e675e5b49c41b3eb6055fad9ec870248a4a65c63e463","target":"graph","created_at":"2026-05-20T00:02:46Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2406.06380/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We present a detailed study of the evolution of the number of connected components in sub-critical multiplicative random graph processes. We consider a model where edges appear independently after an exponential time at rate equal to the product of the sizes of the vertices. We provide an explicit expression for the fluid limit of the number of connected components normalized by its initial value, when the time is smaller than the inverse of the sum of the square of the initial vertex sizes. We also identify the diffusion limit of the rescaled fluctuations around the fluid limit. This is appli","authors_text":"Josu\\'e Corujo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-06-10T15:40:25Z","title":"The number of connected components in sub-critical random graph processes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2406.06380","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:ddec68c0a8052089ecb4b2d8864e3a84192bc7075bfbd2af869b896c667e7a95","target":"record","created_at":"2026-05-20T00:02:46Z","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":"0d4f1fb553cf49395fb9f5324d10fa3db5a27ce9f3ea44d18c86d4ee8df4aa5b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-06-10T15:40:25Z","title_canon_sha256":"142492a4eb73cf6766d8cc6229839b60d5bae9db5241132ada08592ed92e84ff"},"schema_version":"1.0","source":{"id":"2406.06380","kind":"arxiv","version":2}},"canonical_sha256":"a70c4d3d4bff7d354ac75e751a8af1cb12cdde93a9a71413a17392b4a4186400","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a70c4d3d4bff7d354ac75e751a8af1cb12cdde93a9a71413a17392b4a4186400","first_computed_at":"2026-05-20T00:02:46.143262Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:02:46.143262Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QtQG2Qc2YqaB8m0k/xilQ2y1zirh2qN5bgL6SdQH5krb4K2HuSkXraOlJVGXj81rm4b0C9PMFCNVVXbaZMmTDA==","signature_status":"signed_v1","signed_at":"2026-05-20T00:02:46.143897Z","signed_message":"canonical_sha256_bytes"},"source_id":"2406.06380","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ddec68c0a8052089ecb4b2d8864e3a84192bc7075bfbd2af869b896c667e7a95","sha256:b754620ec13fa2201264e675e5b49c41b3eb6055fad9ec870248a4a65c63e463"],"state_sha256":"25dbed5815e4ba899515768c833deea8dabe89ee964adb361994e31047ccf6df"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Igd0+GyIgFWSyvGEU4NrlSNJlv4ZA9iLLZ2Hy3ORF56SoSigZ/t+G5KdXBHEu9n1NXk29NYMNHbLUjQHiZocDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-07T22:04:56.427811Z","bundle_sha256":"347415119ca23aa65413fdd0b859387d2cd6375b0631a4d7c59003fa43e7737b"}}