{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:IYMYZ2CRNTB3GEHUWKMWHRAIPK","short_pith_number":"pith:IYMYZ2CR","canonical_record":{"source":{"id":"1404.2425","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-04-09T10:35:29Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"7cfb616cdf3e9611e7d7e0036800998fcab9511a2eb5927c4e17e161d2a3b64c","abstract_canon_sha256":"33eb73d8b74a4c02f93774eed88f542abf7767daa7d5b672453decea23f01417"},"schema_version":"1.0"},"canonical_sha256":"46198ce8516cc3b310f4b29963c4087a86e70a6bc83d42d6bd32c6919c4d93a3","source":{"kind":"arxiv","id":"1404.2425","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1404.2425","created_at":"2026-05-18T02:54:20Z"},{"alias_kind":"arxiv_version","alias_value":"1404.2425v2","created_at":"2026-05-18T02:54:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1404.2425","created_at":"2026-05-18T02:54:20Z"},{"alias_kind":"pith_short_12","alias_value":"IYMYZ2CRNTB3","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_16","alias_value":"IYMYZ2CRNTB3GEHU","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_8","alias_value":"IYMYZ2CR","created_at":"2026-05-18T12:28:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:IYMYZ2CRNTB3GEHUWKMWHRAIPK","target":"record","payload":{"canonical_record":{"source":{"id":"1404.2425","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-04-09T10:35:29Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"7cfb616cdf3e9611e7d7e0036800998fcab9511a2eb5927c4e17e161d2a3b64c","abstract_canon_sha256":"33eb73d8b74a4c02f93774eed88f542abf7767daa7d5b672453decea23f01417"},"schema_version":"1.0"},"canonical_sha256":"46198ce8516cc3b310f4b29963c4087a86e70a6bc83d42d6bd32c6919c4d93a3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:54:20.703493Z","signature_b64":"8L7DhQQyZCDMJVCa+I7IMSbARSTnLDChUVkk1329wyoaFKm022ZTmhKPh3JUSSCMNGqoCKLGpYFQJTNlafsABA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"46198ce8516cc3b310f4b29963c4087a86e70a6bc83d42d6bd32c6919c4d93a3","last_reissued_at":"2026-05-18T02:54:20.703064Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:54:20.703064Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1404.2425","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":"2026-05-18T02:54:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tA4o6DVzvbsX2iI0TBhcuuSFTcLS3mhHK9LB/MQHcr1ycme9DTr8lCQ9t05NM66St6z4hQny26orgV2QFdcRDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T18:45:20.383346Z"},"content_sha256":"abeb84b8d17b7d28b571ecff52b6f667a67a50fb351c6a7c11951b2380d69fd4","schema_version":"1.0","event_id":"sha256:abeb84b8d17b7d28b571ecff52b6f667a67a50fb351c6a7c11951b2380d69fd4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:IYMYZ2CRNTB3GEHUWKMWHRAIPK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Longest paths in random Apollonian networks and largest $r$-ary subtrees of random $d$-ary recursive trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Abbas Mehrabian, Andrea Collevecchio, Nick Wormald","submitted_at":"2014-04-09T10:35:29Z","abstract_excerpt":"Let $r$ and $d$ be positive integers with $r<d$. Consider a random $d$-ary tree constructed as follows. Start with a single vertex, and in each time-step choose a uniformly random leaf and give it $d$ newly created offspring. Let ${\\mathcal T}_t$ be the tree produced after $t$ steps. We show that there exists a fixed $\\delta<1$ depending on $d$ and $r$ such that almost surely for all large $t$, every $r$-ary subtree of ${\\mathcal T}_t$ has less than $t^{\\delta}$ vertices.\n  The proof involves analysis that also yields a related result. Consider the following iterative construction of a random "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.2425","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":"2026-05-18T02:54:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ztH233kEwtjdmZ9ZCCvtq14f8MQCC6lgvv7qsCePGVxkksPkg8lo75Yck/wGyzS1wntMLwdDM6hL2PVcMuHDCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T18:45:20.383857Z"},"content_sha256":"c3f7f37829274a54d981378a17198c9c46d43b7c23528574169b0c9d7099515e","schema_version":"1.0","event_id":"sha256:c3f7f37829274a54d981378a17198c9c46d43b7c23528574169b0c9d7099515e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IYMYZ2CRNTB3GEHUWKMWHRAIPK/bundle.json","state_url":"https://pith.science/pith/IYMYZ2CRNTB3GEHUWKMWHRAIPK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IYMYZ2CRNTB3GEHUWKMWHRAIPK/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":"2026-05-25T18:45:20Z","links":{"resolver":"https://pith.science/pith/IYMYZ2CRNTB3GEHUWKMWHRAIPK","bundle":"https://pith.science/pith/IYMYZ2CRNTB3GEHUWKMWHRAIPK/bundle.json","state":"https://pith.science/pith/IYMYZ2CRNTB3GEHUWKMWHRAIPK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IYMYZ2CRNTB3GEHUWKMWHRAIPK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:IYMYZ2CRNTB3GEHUWKMWHRAIPK","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":"33eb73d8b74a4c02f93774eed88f542abf7767daa7d5b672453decea23f01417","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-04-09T10:35:29Z","title_canon_sha256":"7cfb616cdf3e9611e7d7e0036800998fcab9511a2eb5927c4e17e161d2a3b64c"},"schema_version":"1.0","source":{"id":"1404.2425","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1404.2425","created_at":"2026-05-18T02:54:20Z"},{"alias_kind":"arxiv_version","alias_value":"1404.2425v2","created_at":"2026-05-18T02:54:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1404.2425","created_at":"2026-05-18T02:54:20Z"},{"alias_kind":"pith_short_12","alias_value":"IYMYZ2CRNTB3","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_16","alias_value":"IYMYZ2CRNTB3GEHU","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_8","alias_value":"IYMYZ2CR","created_at":"2026-05-18T12:28:33Z"}],"graph_snapshots":[{"event_id":"sha256:c3f7f37829274a54d981378a17198c9c46d43b7c23528574169b0c9d7099515e","target":"graph","created_at":"2026-05-18T02:54:20Z","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":"Let $r$ and $d$ be positive integers with $r<d$. Consider a random $d$-ary tree constructed as follows. Start with a single vertex, and in each time-step choose a uniformly random leaf and give it $d$ newly created offspring. Let ${\\mathcal T}_t$ be the tree produced after $t$ steps. We show that there exists a fixed $\\delta<1$ depending on $d$ and $r$ such that almost surely for all large $t$, every $r$-ary subtree of ${\\mathcal T}_t$ has less than $t^{\\delta}$ vertices.\n  The proof involves analysis that also yields a related result. Consider the following iterative construction of a random ","authors_text":"Abbas Mehrabian, Andrea Collevecchio, Nick Wormald","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-04-09T10:35:29Z","title":"Longest paths in random Apollonian networks and largest $r$-ary subtrees of random $d$-ary recursive trees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.2425","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:abeb84b8d17b7d28b571ecff52b6f667a67a50fb351c6a7c11951b2380d69fd4","target":"record","created_at":"2026-05-18T02:54:20Z","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":"33eb73d8b74a4c02f93774eed88f542abf7767daa7d5b672453decea23f01417","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-04-09T10:35:29Z","title_canon_sha256":"7cfb616cdf3e9611e7d7e0036800998fcab9511a2eb5927c4e17e161d2a3b64c"},"schema_version":"1.0","source":{"id":"1404.2425","kind":"arxiv","version":2}},"canonical_sha256":"46198ce8516cc3b310f4b29963c4087a86e70a6bc83d42d6bd32c6919c4d93a3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"46198ce8516cc3b310f4b29963c4087a86e70a6bc83d42d6bd32c6919c4d93a3","first_computed_at":"2026-05-18T02:54:20.703064Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:54:20.703064Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8L7DhQQyZCDMJVCa+I7IMSbARSTnLDChUVkk1329wyoaFKm022ZTmhKPh3JUSSCMNGqoCKLGpYFQJTNlafsABA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:54:20.703493Z","signed_message":"canonical_sha256_bytes"},"source_id":"1404.2425","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:abeb84b8d17b7d28b571ecff52b6f667a67a50fb351c6a7c11951b2380d69fd4","sha256:c3f7f37829274a54d981378a17198c9c46d43b7c23528574169b0c9d7099515e"],"state_sha256":"268d5b0fb3194f61afede22a740de29cb0d034a6782f5929d883b5de41c04ee8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hijf5R8PF8A2FII0L72fUTPdUXdvSgpGBlwAnnMJZbvEPc5F+mFPzRr9HLa/2mPFIIE3zlAOBB/AXLEjg+RaCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T18:45:20.386859Z","bundle_sha256":"c3e5075ec821a11c17c181ece9818939da8c5fae16ab569baa6e933854c01a93"}}