{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:56KML5TUD2BKK476HALDKJD34T","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":"900eea0ea02f620e6830039ba4d86f0633d3d21362eae3214b8b6eaf34f90857","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-03-21T09:52:09Z","title_canon_sha256":"cce32dfc70974ef27944f0e00967deed3deb64580bee084b058e14826e10127c"},"schema_version":"1.0","source":{"id":"1303.5213","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.5213","created_at":"2026-05-18T03:30:13Z"},{"alias_kind":"arxiv_version","alias_value":"1303.5213v1","created_at":"2026-05-18T03:30:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.5213","created_at":"2026-05-18T03:30:13Z"},{"alias_kind":"pith_short_12","alias_value":"56KML5TUD2BK","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_16","alias_value":"56KML5TUD2BKK476","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_8","alias_value":"56KML5TU","created_at":"2026-05-18T12:27:34Z"}],"graph_snapshots":[{"event_id":"sha256:a55dcd7cd11d63c26a7d30f7d67a5d23b6fec88651702e658424f1933b85291b","target":"graph","created_at":"2026-05-18T03:30:13Z","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":"We consider the following iterative construction of a random planar triangulation. Start with a triangle embedded in the plane. In each step, choose a bounded face uniformly at random, add a vertex inside that face and join it to the vertices of the face. After n-3 steps, we obtain a random triangulated plane graph with n vertices, which is called a Random Apollonian Network (RAN). We show that asymptotically almost surely (a.a.s.) every path in a RAN has length o(n), refuting a conjecture of Frieze and Tsourakakis. We also show that a RAN always has a path of length (2n-5)^{log 2/log 3}, and ","authors_text":"Abbas Mehrabian, Cristiane M. Sato, Ehsan Ebrahimzadeh, Jonathan Zung, Linda Farczadi, Nick Wormald, Pu Gao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-03-21T09:52:09Z","title":"On the Longest Paths and the Diameter in Random Apollonian Networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.5213","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":"sha256:43b883ccea417abd9fbeb5e4dcc1922e8aec2b094997a302ee3d8f75d61272e2","target":"record","created_at":"2026-05-18T03:30:13Z","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":"900eea0ea02f620e6830039ba4d86f0633d3d21362eae3214b8b6eaf34f90857","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-03-21T09:52:09Z","title_canon_sha256":"cce32dfc70974ef27944f0e00967deed3deb64580bee084b058e14826e10127c"},"schema_version":"1.0","source":{"id":"1303.5213","kind":"arxiv","version":1}},"canonical_sha256":"ef94c5f6741e82a573fe381635247be4e9d3e1972b3d6f02ee7980864103db54","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ef94c5f6741e82a573fe381635247be4e9d3e1972b3d6f02ee7980864103db54","first_computed_at":"2026-05-18T03:30:13.601589Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:30:13.601589Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Uj3bNbfRj6QvXsDSQw5fTFPu1XJHLMnur1g0T3XLNbkeuZSGNtG3zXVurv0Y8Ck6eAqyXSuXFoMG+NNVNNSNDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:30:13.602363Z","signed_message":"canonical_sha256_bytes"},"source_id":"1303.5213","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:43b883ccea417abd9fbeb5e4dcc1922e8aec2b094997a302ee3d8f75d61272e2","sha256:a55dcd7cd11d63c26a7d30f7d67a5d23b6fec88651702e658424f1933b85291b"],"state_sha256":"18de683538e94e4177f332f11e57d6b649b507d2b856e07dca6d920afff47d1d"}