{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:EWYRD7JTZKC45VQECDKVI7LVOS","short_pith_number":"pith:EWYRD7JT","schema_version":"1.0","canonical_sha256":"25b111fd33ca85ced60410d5547d75748b6b61706d07d12df78a70e18c202eba","source":{"kind":"arxiv","id":"1609.00290","version":1},"attestation_state":"computed","paper":{"title":"Counting strongly connected $(k_1,k_2)$-directed cores","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Boris Pittel","submitted_at":"2016-09-01T15:51:56Z","abstract_excerpt":"Consider the set of all digraphs on $[N]$ with $M$ edges, whose minimum in-degree and minimum out-degree are at least $k_1$ and $k_2$ respectively. For $k:=\\min\\{k_1,k_2\\}\\ge 2$ and $M/N>\\max\\{k_1,k_2\\}$, $M=\\Theta(N)$, we show that, among those digraphs, the fraction of $k$-strongly connected digraphs is $1-O\\bigl(N^{-(k-1)})$. Earlier with Dan Poole we identified a sharp edge-density threshold $c^*(k_1,k_2)$ for birth of a giant $(k_1,k_2)$-core in the random digraph $D(n,m=[cn])$. Combining the claims, for $c>c^*(k_1,k_2)$ with probability $1-O\\bigl(N^{-(k-1)})$ the giant $(k_1,k_2)$-core e"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1609.00290","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-01T15:51:56Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"10b127ebd5c22668fd2e8ef8be4ea38709d4d86aa7010e7225d085430165de4d","abstract_canon_sha256":"8e513553276a4ac1d6ead51f36c27554386403b3effd03f67541ec37caa9d865"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:06:25.210612Z","signature_b64":"u8KzHcupRH+OR6BoRNmPPPmI5a+uApTvyv/DNElxmrukxo8kiXp0aiGy+BoN+NDyKLbZUejCEGeXMVOwqDPiDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"25b111fd33ca85ced60410d5547d75748b6b61706d07d12df78a70e18c202eba","last_reissued_at":"2026-05-18T01:06:25.210047Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:06:25.210047Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Counting strongly connected $(k_1,k_2)$-directed cores","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Boris Pittel","submitted_at":"2016-09-01T15:51:56Z","abstract_excerpt":"Consider the set of all digraphs on $[N]$ with $M$ edges, whose minimum in-degree and minimum out-degree are at least $k_1$ and $k_2$ respectively. For $k:=\\min\\{k_1,k_2\\}\\ge 2$ and $M/N>\\max\\{k_1,k_2\\}$, $M=\\Theta(N)$, we show that, among those digraphs, the fraction of $k$-strongly connected digraphs is $1-O\\bigl(N^{-(k-1)})$. Earlier with Dan Poole we identified a sharp edge-density threshold $c^*(k_1,k_2)$ for birth of a giant $(k_1,k_2)$-core in the random digraph $D(n,m=[cn])$. Combining the claims, for $c>c^*(k_1,k_2)$ with probability $1-O\\bigl(N^{-(k-1)})$ the giant $(k_1,k_2)$-core e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.00290","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1609.00290","created_at":"2026-05-18T01:06:25.210133+00:00"},{"alias_kind":"arxiv_version","alias_value":"1609.00290v1","created_at":"2026-05-18T01:06:25.210133+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.00290","created_at":"2026-05-18T01:06:25.210133+00:00"},{"alias_kind":"pith_short_12","alias_value":"EWYRD7JTZKC4","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_16","alias_value":"EWYRD7JTZKC45VQE","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_8","alias_value":"EWYRD7JT","created_at":"2026-05-18T12:30:15.759754+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/EWYRD7JTZKC45VQECDKVI7LVOS","json":"https://pith.science/pith/EWYRD7JTZKC45VQECDKVI7LVOS.json","graph_json":"https://pith.science/api/pith-number/EWYRD7JTZKC45VQECDKVI7LVOS/graph.json","events_json":"https://pith.science/api/pith-number/EWYRD7JTZKC45VQECDKVI7LVOS/events.json","paper":"https://pith.science/paper/EWYRD7JT"},"agent_actions":{"view_html":"https://pith.science/pith/EWYRD7JTZKC45VQECDKVI7LVOS","download_json":"https://pith.science/pith/EWYRD7JTZKC45VQECDKVI7LVOS.json","view_paper":"https://pith.science/paper/EWYRD7JT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1609.00290&json=true","fetch_graph":"https://pith.science/api/pith-number/EWYRD7JTZKC45VQECDKVI7LVOS/graph.json","fetch_events":"https://pith.science/api/pith-number/EWYRD7JTZKC45VQECDKVI7LVOS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EWYRD7JTZKC45VQECDKVI7LVOS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EWYRD7JTZKC45VQECDKVI7LVOS/action/storage_attestation","attest_author":"https://pith.science/pith/EWYRD7JTZKC45VQECDKVI7LVOS/action/author_attestation","sign_citation":"https://pith.science/pith/EWYRD7JTZKC45VQECDKVI7LVOS/action/citation_signature","submit_replication":"https://pith.science/pith/EWYRD7JTZKC45VQECDKVI7LVOS/action/replication_record"}},"created_at":"2026-05-18T01:06:25.210133+00:00","updated_at":"2026-05-18T01:06:25.210133+00:00"}