{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:WSBJEJVD246R5QYONXMEHWJBEZ","short_pith_number":"pith:WSBJEJVD","canonical_record":{"source":{"id":"0906.0152","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-05-31T12:51:46Z","cross_cats_sorted":[],"title_canon_sha256":"8be1edbb07a2130fc0014750386900ff9b1d8778fb7e334ad4f9f7807b0025f4","abstract_canon_sha256":"6fb429bd6f6222bbea72e477996ee979631dbe85d28db1bc1fb53c3299ea187f"},"schema_version":"1.0"},"canonical_sha256":"b4829226a3d73d1ec30e6dd843d921266d5b569dc9bc03dec93bd1894d38ed65","source":{"kind":"arxiv","id":"0906.0152","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0906.0152","created_at":"2026-05-18T02:13:30Z"},{"alias_kind":"arxiv_version","alias_value":"0906.0152v1","created_at":"2026-05-18T02:13:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0906.0152","created_at":"2026-05-18T02:13:30Z"},{"alias_kind":"pith_short_12","alias_value":"WSBJEJVD246R","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_16","alias_value":"WSBJEJVD246R5QYO","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_8","alias_value":"WSBJEJVD","created_at":"2026-05-18T12:26:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:WSBJEJVD246R5QYONXMEHWJBEZ","target":"record","payload":{"canonical_record":{"source":{"id":"0906.0152","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-05-31T12:51:46Z","cross_cats_sorted":[],"title_canon_sha256":"8be1edbb07a2130fc0014750386900ff9b1d8778fb7e334ad4f9f7807b0025f4","abstract_canon_sha256":"6fb429bd6f6222bbea72e477996ee979631dbe85d28db1bc1fb53c3299ea187f"},"schema_version":"1.0"},"canonical_sha256":"b4829226a3d73d1ec30e6dd843d921266d5b569dc9bc03dec93bd1894d38ed65","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:13:30.283220Z","signature_b64":"76Z1jxGEWdcg7k6D+9LLOUAVU4Hsft1aEPG1MQXkYJV/mOwBePs+Ar8ZSPJb0stVKLXtNZQeDbqaUJhTALcJAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b4829226a3d73d1ec30e6dd843d921266d5b569dc9bc03dec93bd1894d38ed65","last_reissued_at":"2026-05-18T02:13:30.282518Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:13:30.282518Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0906.0152","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:13:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dvL3GbSCIAGBOl2qiRCRNBmZC6bX1DNEOqiKOPSY+xL+tlzGA2VPfwdwCl/38bTxQUFapBElvpQ3NTv7R2VaCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T11:58:37.037831Z"},"content_sha256":"9bdbdc937b4c35f4174e52cee9b4c15549f78aa215c70a24aac6683b217bd68e","schema_version":"1.0","event_id":"sha256:9bdbdc937b4c35f4174e52cee9b4c15549f78aa215c70a24aac6683b217bd68e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:WSBJEJVD246R5QYONXMEHWJBEZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Long and short paths in uniform random recursive dags","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Luc Devroye, Svante Janson","submitted_at":"2009-05-31T12:51:46Z","abstract_excerpt":"In a uniform random recursive k-dag, there is a root, 0, and each node in turn, from 1 to n, chooses k uniform random parents from among the nodes of smaller index. If S_n is the shortest path distance from node n to the root, then we determine the constant \\sigma such that S_n/log(n) tends to \\sigma in probability as n tends to infinity. We also show that max_{1 \\le i \\le n} S_i/log(n) tends to \\sigma in probability."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.0152","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:13:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rAgw85KpoJKhxAkz36m+Dhq+JN1KJfy4Tjb2NIC1F7jgfsR/xf5cj8KCst8kKFp7yREB6oq5llCJ736KIM10DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T11:58:37.038180Z"},"content_sha256":"99e54611fef121abfa45467fa367b8848fe006f959ee8e50ad476445c9704709","schema_version":"1.0","event_id":"sha256:99e54611fef121abfa45467fa367b8848fe006f959ee8e50ad476445c9704709"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WSBJEJVD246R5QYONXMEHWJBEZ/bundle.json","state_url":"https://pith.science/pith/WSBJEJVD246R5QYONXMEHWJBEZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WSBJEJVD246R5QYONXMEHWJBEZ/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":"2026-05-26T11:58:37Z","links":{"resolver":"https://pith.science/pith/WSBJEJVD246R5QYONXMEHWJBEZ","bundle":"https://pith.science/pith/WSBJEJVD246R5QYONXMEHWJBEZ/bundle.json","state":"https://pith.science/pith/WSBJEJVD246R5QYONXMEHWJBEZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WSBJEJVD246R5QYONXMEHWJBEZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:WSBJEJVD246R5QYONXMEHWJBEZ","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":"6fb429bd6f6222bbea72e477996ee979631dbe85d28db1bc1fb53c3299ea187f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-05-31T12:51:46Z","title_canon_sha256":"8be1edbb07a2130fc0014750386900ff9b1d8778fb7e334ad4f9f7807b0025f4"},"schema_version":"1.0","source":{"id":"0906.0152","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0906.0152","created_at":"2026-05-18T02:13:30Z"},{"alias_kind":"arxiv_version","alias_value":"0906.0152v1","created_at":"2026-05-18T02:13:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0906.0152","created_at":"2026-05-18T02:13:30Z"},{"alias_kind":"pith_short_12","alias_value":"WSBJEJVD246R","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_16","alias_value":"WSBJEJVD246R5QYO","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_8","alias_value":"WSBJEJVD","created_at":"2026-05-18T12:26:02Z"}],"graph_snapshots":[{"event_id":"sha256:99e54611fef121abfa45467fa367b8848fe006f959ee8e50ad476445c9704709","target":"graph","created_at":"2026-05-18T02:13:30Z","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":"In a uniform random recursive k-dag, there is a root, 0, and each node in turn, from 1 to n, chooses k uniform random parents from among the nodes of smaller index. If S_n is the shortest path distance from node n to the root, then we determine the constant \\sigma such that S_n/log(n) tends to \\sigma in probability as n tends to infinity. We also show that max_{1 \\le i \\le n} S_i/log(n) tends to \\sigma in probability.","authors_text":"Luc Devroye, Svante Janson","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-05-31T12:51:46Z","title":"Long and short paths in uniform random recursive dags"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.0152","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":"sha256:9bdbdc937b4c35f4174e52cee9b4c15549f78aa215c70a24aac6683b217bd68e","target":"record","created_at":"2026-05-18T02:13:30Z","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":"6fb429bd6f6222bbea72e477996ee979631dbe85d28db1bc1fb53c3299ea187f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-05-31T12:51:46Z","title_canon_sha256":"8be1edbb07a2130fc0014750386900ff9b1d8778fb7e334ad4f9f7807b0025f4"},"schema_version":"1.0","source":{"id":"0906.0152","kind":"arxiv","version":1}},"canonical_sha256":"b4829226a3d73d1ec30e6dd843d921266d5b569dc9bc03dec93bd1894d38ed65","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b4829226a3d73d1ec30e6dd843d921266d5b569dc9bc03dec93bd1894d38ed65","first_computed_at":"2026-05-18T02:13:30.282518Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:13:30.282518Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"76Z1jxGEWdcg7k6D+9LLOUAVU4Hsft1aEPG1MQXkYJV/mOwBePs+Ar8ZSPJb0stVKLXtNZQeDbqaUJhTALcJAA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:13:30.283220Z","signed_message":"canonical_sha256_bytes"},"source_id":"0906.0152","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9bdbdc937b4c35f4174e52cee9b4c15549f78aa215c70a24aac6683b217bd68e","sha256:99e54611fef121abfa45467fa367b8848fe006f959ee8e50ad476445c9704709"],"state_sha256":"c9bc7b103cb17569d3d8bf72ab21405a0df905914730a0d9575822ca0179bc38"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"v5h1IV5n0X5uTtA+YT/BS+W6FtaJInIe+GbyYZqCnMs9vjOgxjM/u00CEDfcC+S39BpKm891tFvSojltDyLUDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T11:58:37.041088Z","bundle_sha256":"f69fdcaaa9dac1768e80f1cc57124451bcf11c7c37df4245d666fff59dda2925"}}