{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:ON6OAAVQLGOUAVHOTLJOU2UEQV","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":"5de0b331c286442ea484fd6d6b2fa8721f860eb4f5771791a5052bb7afeb7036","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.PR","submitted_at":"2024-12-18T15:54:37Z","title_canon_sha256":"aa4a79ad95e15b86848acd6ec17e3faa08e11e6c6ceb879eda173c5a206d9944"},"schema_version":"1.0","source":{"id":"2412.13975","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.13975","created_at":"2026-07-05T09:51:11Z"},{"alias_kind":"arxiv_version","alias_value":"2412.13975v1","created_at":"2026-07-05T09:51:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.13975","created_at":"2026-07-05T09:51:11Z"},{"alias_kind":"pith_short_12","alias_value":"ON6OAAVQLGOU","created_at":"2026-07-05T09:51:11Z"},{"alias_kind":"pith_short_16","alias_value":"ON6OAAVQLGOUAVHO","created_at":"2026-07-05T09:51:11Z"},{"alias_kind":"pith_short_8","alias_value":"ON6OAAVQ","created_at":"2026-07-05T09:51:11Z"}],"graph_snapshots":[{"event_id":"sha256:a5a3b525da59769ef609b953c0448185e87d282b08095f577925e15a72409355","target":"graph","created_at":"2026-07-05T09:51:11Z","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/2412.13975/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the number $X^{(n)}$ of vertices that can be reached from the last added vertex $n$ via a directed path (the descendants) in the standard preferential attachment graph. In this model, vertices are sequentially added, each born with outdegree $m\\ge 2$; the endpoint of each outgoing edge is chosen among previously added vertices with probability proportional to the current degree of the vertex plus some number $\\rho$.\n  We show that $X^{(n)}/n^\\nu$ converges in distribution as $n\\to\\infty$, where $\\nu$ depends on both $m$ and $\\rho$, and the limiting distribution is given by a product o","authors_text":"Svante Janson, Tiffany Y. Y. Lo","cross_cats":["math.CO"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.PR","submitted_at":"2024-12-18T15:54:37Z","title":"The number of descendants in a preferential attachment graph"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.13975","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":"sha256:2654c5463d88b4e9187c80ce4de86e06cba7461323eb252bd3a023d664f86b64","target":"record","created_at":"2026-07-05T09:51:11Z","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":"5de0b331c286442ea484fd6d6b2fa8721f860eb4f5771791a5052bb7afeb7036","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.PR","submitted_at":"2024-12-18T15:54:37Z","title_canon_sha256":"aa4a79ad95e15b86848acd6ec17e3faa08e11e6c6ceb879eda173c5a206d9944"},"schema_version":"1.0","source":{"id":"2412.13975","kind":"arxiv","version":1}},"canonical_sha256":"737ce002b0599d4054ee9ad2ea6a848553515dd157d2c8a958b9cb18e25dd550","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"737ce002b0599d4054ee9ad2ea6a848553515dd157d2c8a958b9cb18e25dd550","first_computed_at":"2026-07-05T09:51:11.683841Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:51:11.683841Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IDjyH3hrNpYpPTYNQviMN/KcFHHtKSXRjnb3FMTWGDh+8YMuX0a1IkdtiBlNpDpvkfEWKAMfaCIGfpuivQTLDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T09:51:11.684329Z","signed_message":"canonical_sha256_bytes"},"source_id":"2412.13975","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2654c5463d88b4e9187c80ce4de86e06cba7461323eb252bd3a023d664f86b64","sha256:a5a3b525da59769ef609b953c0448185e87d282b08095f577925e15a72409355"],"state_sha256":"84459c8da2d299ef21db6118bc12a27262522f09ae42a341b416c55a8a253f1f"}