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If d=d(n) tends to infinity with n, the stationary distribution is asymptotically uniform whp.\n  Using this result we prove that, for d>1, whp the cover time of D_{n,p} is asymptotic to d\\log (d/(d-1))n\\log n. If d=d(n) tends to infinity with n, then the cover time is asymptotic to n\\log n."},"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":"1103.4317","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-03-22T17:24:54Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"f8c94b2f1e72670cbd57a8c3368bed30537233095a2991106aaa52b0eed2bd77","abstract_canon_sha256":"625dd8a048064d9a0a75ab3d4328636e5c66e3fc6d22f265db4363ade7859bf0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:26:12.578633Z","signature_b64":"x0HR5QGYc89v5nef0uo0hgnLizDVBQRVDzVujp5Y1FqbuqyeWGIBs/DmmpRW0lKN+TWOjLthAfVF4z1dJhDXBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d47dee43e9d21fb09b3a841c15b23f1cd3559c5b81a8804f5c131ceafb76af2c","last_reissued_at":"2026-05-18T04:26:12.577992Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:26:12.577992Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Stationary distribution and cover time of random walks on random digraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Alan Frieze, Colin Cooper","submitted_at":"2011-03-22T17:24:54Z","abstract_excerpt":"We study properties of a simple random walk on the random digraph D_{n,p} when np={d\\log n},\\; d>1.\n  We prove that whp the stationary probability pi_v of a vertex v is asymptotic to deg^-(v)/m where deg^-(v) is the in-degree of v and m=n(n-1)p is the expected number of edges of D_{n,p}. If d=d(n) tends to infinity with n, the stationary distribution is asymptotically uniform whp.\n  Using this result we prove that, for d>1, whp the cover time of D_{n,p} is asymptotic to d\\log (d/(d-1))n\\log n. If d=d(n) tends to infinity with n, then the cover time is asymptotic to n\\log n."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.4317","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":"1103.4317","created_at":"2026-05-18T04:26:12.578089+00:00"},{"alias_kind":"arxiv_version","alias_value":"1103.4317v1","created_at":"2026-05-18T04:26:12.578089+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.4317","created_at":"2026-05-18T04:26:12.578089+00:00"},{"alias_kind":"pith_short_12","alias_value":"2R664Q7J2IP3","created_at":"2026-05-18T12:26:18.847500+00:00"},{"alias_kind":"pith_short_16","alias_value":"2R664Q7J2IP3BGZ2","created_at":"2026-05-18T12:26:18.847500+00:00"},{"alias_kind":"pith_short_8","alias_value":"2R664Q7J","created_at":"2026-05-18T12:26:18.847500+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/2R664Q7J2IP3BGZ2QQOBLMR7DT","json":"https://pith.science/pith/2R664Q7J2IP3BGZ2QQOBLMR7DT.json","graph_json":"https://pith.science/api/pith-number/2R664Q7J2IP3BGZ2QQOBLMR7DT/graph.json","events_json":"https://pith.science/api/pith-number/2R664Q7J2IP3BGZ2QQOBLMR7DT/events.json","paper":"https://pith.science/paper/2R664Q7J"},"agent_actions":{"view_html":"https://pith.science/pith/2R664Q7J2IP3BGZ2QQOBLMR7DT","download_json":"https://pith.science/pith/2R664Q7J2IP3BGZ2QQOBLMR7DT.json","view_paper":"https://pith.science/paper/2R664Q7J","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1103.4317&json=true","fetch_graph":"https://pith.science/api/pith-number/2R664Q7J2IP3BGZ2QQOBLMR7DT/graph.json","fetch_events":"https://pith.science/api/pith-number/2R664Q7J2IP3BGZ2QQOBLMR7DT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2R664Q7J2IP3BGZ2QQOBLMR7DT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2R664Q7J2IP3BGZ2QQOBLMR7DT/action/storage_attestation","attest_author":"https://pith.science/pith/2R664Q7J2IP3BGZ2QQOBLMR7DT/action/author_attestation","sign_citation":"https://pith.science/pith/2R664Q7J2IP3BGZ2QQOBLMR7DT/action/citation_signature","submit_replication":"https://pith.science/pith/2R664Q7J2IP3BGZ2QQOBLMR7DT/action/replication_record"}},"created_at":"2026-05-18T04:26:12.578089+00:00","updated_at":"2026-05-18T04:26:12.578089+00:00"}