{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:RZ7SZH7C7QAMWQ4ZLDAHYW3MVL","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":"49320e6cebee6f7cdbea1e1966c0dfd21619632aee27ddbcbb67c4e4bcf3d6a2","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-05-15T21:50:28Z","title_canon_sha256":"1263bef52bcd319ec3c45d00bd568b15eb726c13c99a0ea16ed2f3185a985207"},"schema_version":"1.0","source":{"id":"1405.4022","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.4022","created_at":"2026-05-18T02:03:55Z"},{"alias_kind":"arxiv_version","alias_value":"1405.4022v2","created_at":"2026-05-18T02:03:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.4022","created_at":"2026-05-18T02:03:55Z"},{"alias_kind":"pith_short_12","alias_value":"RZ7SZH7C7QAM","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_16","alias_value":"RZ7SZH7C7QAMWQ4Z","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_8","alias_value":"RZ7SZH7C","created_at":"2026-05-18T12:28:46Z"}],"graph_snapshots":[{"event_id":"sha256:c8b543a6486d79860a71a999f8e876fa40da7e2a7efeb1efe9b7b8669880da71","target":"graph","created_at":"2026-05-18T02:03:55Z","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":"Two models of a random digraph on $n$ vertices, $D(n,\\text{Prob}(\\text{arc})=p)$ and $D(n,\\text{number of arcs}=m)$ are studied. In 1990, Karp for $D(n,p)$ and independently T. \\L uczak for $D(n,m=cn)$ proved that for $c>1$, with probability tending to 1, there is an unique strong component of size of order $n$. Karp showed, in fact, that the giant component has likely size asymptotic to $n\\theta^2$, where $\\theta=\\theta(c)$ is the unique positive root of $1-\\theta=e^{-c \\theta}$. In this paper we prove that, for both random digraphs, the joint distribution of the number of vertices and number","authors_text":"Boris Pittel, Daniel Poole","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-05-15T21:50:28Z","title":"Asymptotic distribution of the numbers of vertices and arcs of the giant strong component in sparse random digraphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.4022","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":"sha256:22a97fdad32a3850359e00d5400564b9ff7944195d2fb50ee40eb363465cb3f2","target":"record","created_at":"2026-05-18T02:03:55Z","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":"49320e6cebee6f7cdbea1e1966c0dfd21619632aee27ddbcbb67c4e4bcf3d6a2","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-05-15T21:50:28Z","title_canon_sha256":"1263bef52bcd319ec3c45d00bd568b15eb726c13c99a0ea16ed2f3185a985207"},"schema_version":"1.0","source":{"id":"1405.4022","kind":"arxiv","version":2}},"canonical_sha256":"8e7f2c9fe2fc00cb439958c07c5b6caafd267fcb7da66a5009425352ac136c08","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8e7f2c9fe2fc00cb439958c07c5b6caafd267fcb7da66a5009425352ac136c08","first_computed_at":"2026-05-18T02:03:55.595485Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:03:55.595485Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zFrqgjiKtP7NGbsj3eSIYiI8AE4+YOemX+lxp84Y57EhJWMXHFGKvlvlR9czJJUTzb4gGyR9VeFFeBkipWJkAw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:03:55.596282Z","signed_message":"canonical_sha256_bytes"},"source_id":"1405.4022","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:22a97fdad32a3850359e00d5400564b9ff7944195d2fb50ee40eb363465cb3f2","sha256:c8b543a6486d79860a71a999f8e876fa40da7e2a7efeb1efe9b7b8669880da71"],"state_sha256":"92db5e5cc7e95f01ae966703a530d87cb1495d8d0f6245555896c242832be2b1"}