{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:USUOWQLZX4KBJKCFWKDJCFIUHR","short_pith_number":"pith:USUOWQLZ","canonical_record":{"source":{"id":"0810.4908","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-10-27T19:15:31Z","cross_cats_sorted":["cond-mat.stat-mech","math.CO"],"title_canon_sha256":"9b4cb4d4e08a9844d77fe55a746b3a2119770458e76e70c4682cc4afe772df94","abstract_canon_sha256":"5f03c19597be6aa098207c07e34052c116e08d64a579c245ad880d2c8666b9d6"},"schema_version":"1.0"},"canonical_sha256":"a4a8eb4179bf1414a845b2869115143c5721d18ddb5befb117a7e84df2401133","source":{"kind":"arxiv","id":"0810.4908","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0810.4908","created_at":"2026-05-18T03:54:06Z"},{"alias_kind":"arxiv_version","alias_value":"0810.4908v2","created_at":"2026-05-18T03:54:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0810.4908","created_at":"2026-05-18T03:54:06Z"},{"alias_kind":"pith_short_12","alias_value":"USUOWQLZX4KB","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"USUOWQLZX4KBJKCF","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"USUOWQLZ","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:USUOWQLZX4KBJKCFWKDJCFIUHR","target":"record","payload":{"canonical_record":{"source":{"id":"0810.4908","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-10-27T19:15:31Z","cross_cats_sorted":["cond-mat.stat-mech","math.CO"],"title_canon_sha256":"9b4cb4d4e08a9844d77fe55a746b3a2119770458e76e70c4682cc4afe772df94","abstract_canon_sha256":"5f03c19597be6aa098207c07e34052c116e08d64a579c245ad880d2c8666b9d6"},"schema_version":"1.0"},"canonical_sha256":"a4a8eb4179bf1414a845b2869115143c5721d18ddb5befb117a7e84df2401133","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:54:06.392326Z","signature_b64":"GhfgYpavvL1/3nvj7vOykC/OgCsJee3SrvFoObUK/HhVKZa3eRMkb8o2P6mKi1KmbjT5g13opiHrME+K9TonBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a4a8eb4179bf1414a845b2869115143c5721d18ddb5befb117a7e84df2401133","last_reissued_at":"2026-05-18T03:54:06.391782Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:54:06.391782Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0810.4908","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:54:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"i3x1fQg5HN4P73KednNT0EpyAvBQf6RtKosbL9KvZTSASzwD/R1NzhKOoDPJZhfRteFTovs02bNf+5ib5ow8BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-18T22:51:05.563673Z"},"content_sha256":"2cf4800e775d61ed774cd2e7847bd96c007b4c4b371e9182551168c29f1a12d1","schema_version":"1.0","event_id":"sha256:2cf4800e775d61ed774cd2e7847bd96c007b4c4b371e9182551168c29f1a12d1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:USUOWQLZX4KBJKCFWKDJCFIUHR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A sharp threshold for minimum bounded-depth and bounded-diameter spanning trees and Steiner trees in random networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.stat-mech","math.CO"],"primary_cat":"math.PR","authors_text":"Abraham D. Flaxman, David B. Wilson, Omer Angel","submitted_at":"2008-10-27T19:15:31Z","abstract_excerpt":"In the complete graph on n vertices, when each edge has a weight which is an exponential random variable, Frieze proved that the minimum spanning tree has weight tending to zeta(3)=1/1^3+1/2^3+1/3^3+... as n goes to infinity. We consider spanning trees constrained to have depth bounded by k from a specified root. We prove that if k > log_2 log n+omega(1), where omega(1) is any function going to infinity with n, then the minimum bounded-depth spanning tree still has weight tending to zeta(3) as n -> infinity, and that if k < log_2 log n, then the weight is doubly-exponentially large in log_2 lo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0810.4908","kind":"arxiv","version":2},"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":"2026-05-18T03:54:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"46MHHTGsD1KjCKg9sbOmMndWoY0+ISShy6SMjQu93p1sX980JfwlLRGnatEGfdQrSl+MU7C+aGXnIMjRuLfjDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-18T22:51:05.564363Z"},"content_sha256":"e50fd74da3b98a930d8753e060495ee95beac28d4fe27a9d080bb7c582009063","schema_version":"1.0","event_id":"sha256:e50fd74da3b98a930d8753e060495ee95beac28d4fe27a9d080bb7c582009063"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/USUOWQLZX4KBJKCFWKDJCFIUHR/bundle.json","state_url":"https://pith.science/pith/USUOWQLZX4KBJKCFWKDJCFIUHR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/USUOWQLZX4KBJKCFWKDJCFIUHR/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":"2026-05-18T22:51:05Z","links":{"resolver":"https://pith.science/pith/USUOWQLZX4KBJKCFWKDJCFIUHR","bundle":"https://pith.science/pith/USUOWQLZX4KBJKCFWKDJCFIUHR/bundle.json","state":"https://pith.science/pith/USUOWQLZX4KBJKCFWKDJCFIUHR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/USUOWQLZX4KBJKCFWKDJCFIUHR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:USUOWQLZX4KBJKCFWKDJCFIUHR","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":"5f03c19597be6aa098207c07e34052c116e08d64a579c245ad880d2c8666b9d6","cross_cats_sorted":["cond-mat.stat-mech","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-10-27T19:15:31Z","title_canon_sha256":"9b4cb4d4e08a9844d77fe55a746b3a2119770458e76e70c4682cc4afe772df94"},"schema_version":"1.0","source":{"id":"0810.4908","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0810.4908","created_at":"2026-05-18T03:54:06Z"},{"alias_kind":"arxiv_version","alias_value":"0810.4908v2","created_at":"2026-05-18T03:54:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0810.4908","created_at":"2026-05-18T03:54:06Z"},{"alias_kind":"pith_short_12","alias_value":"USUOWQLZX4KB","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"USUOWQLZX4KBJKCF","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"USUOWQLZ","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:e50fd74da3b98a930d8753e060495ee95beac28d4fe27a9d080bb7c582009063","target":"graph","created_at":"2026-05-18T03:54:06Z","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 the complete graph on n vertices, when each edge has a weight which is an exponential random variable, Frieze proved that the minimum spanning tree has weight tending to zeta(3)=1/1^3+1/2^3+1/3^3+... as n goes to infinity. We consider spanning trees constrained to have depth bounded by k from a specified root. We prove that if k > log_2 log n+omega(1), where omega(1) is any function going to infinity with n, then the minimum bounded-depth spanning tree still has weight tending to zeta(3) as n -> infinity, and that if k < log_2 log n, then the weight is doubly-exponentially large in log_2 lo","authors_text":"Abraham D. Flaxman, David B. Wilson, Omer Angel","cross_cats":["cond-mat.stat-mech","math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-10-27T19:15:31Z","title":"A sharp threshold for minimum bounded-depth and bounded-diameter spanning trees and Steiner trees in random networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0810.4908","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":"sha256:2cf4800e775d61ed774cd2e7847bd96c007b4c4b371e9182551168c29f1a12d1","target":"record","created_at":"2026-05-18T03:54:06Z","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":"5f03c19597be6aa098207c07e34052c116e08d64a579c245ad880d2c8666b9d6","cross_cats_sorted":["cond-mat.stat-mech","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-10-27T19:15:31Z","title_canon_sha256":"9b4cb4d4e08a9844d77fe55a746b3a2119770458e76e70c4682cc4afe772df94"},"schema_version":"1.0","source":{"id":"0810.4908","kind":"arxiv","version":2}},"canonical_sha256":"a4a8eb4179bf1414a845b2869115143c5721d18ddb5befb117a7e84df2401133","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a4a8eb4179bf1414a845b2869115143c5721d18ddb5befb117a7e84df2401133","first_computed_at":"2026-05-18T03:54:06.391782Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:54:06.391782Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GhfgYpavvL1/3nvj7vOykC/OgCsJee3SrvFoObUK/HhVKZa3eRMkb8o2P6mKi1KmbjT5g13opiHrME+K9TonBw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:54:06.392326Z","signed_message":"canonical_sha256_bytes"},"source_id":"0810.4908","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2cf4800e775d61ed774cd2e7847bd96c007b4c4b371e9182551168c29f1a12d1","sha256:e50fd74da3b98a930d8753e060495ee95beac28d4fe27a9d080bb7c582009063"],"state_sha256":"d35398bf819543bdde3354a843b8c51d6f0e1e853824200fa88e28f00b504118"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bi+C9J5VoLre+3gVUheV8x2xE1z0/O7z8YcU2qBXYLFxZlJbXwDF7SKTm9HQ4JwORJUZ7XwLqdBImXLvTLqlDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-18T22:51:05.566672Z","bundle_sha256":"fc5e1811d6b79f209ec941f6b85a8a3e8f61ecef0d2dd3844afa70fe31a3ed0d"}}