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Specifically, let $k$ be a constant and $\\Delta_1 \\geq \\Delta_2 \\geq .. \\geq \\Delta_k$ be the degrees of the $k$ highest degree vertices. We prove that at time $t$, for any function $f$ with $f(t) \\rightarrow +\\infty$ as $t \\rightarrow +\\infty$, $\\frac{t^{1/2}}{f(t)} \\leq \\Delta_1 \\leq f(t)t^{1/2}$ and for $i=2,...,k=O(1)$, $\\frac{t^{1/2}}{f(t)} \\leq \\Delta_i \\leq \\Delta_{i-1} - \\frac{t^{1/2}}{f(t)}$ with high probability "},"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":"1104.5259","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.SI","submitted_at":"2011-04-27T21:55:21Z","cross_cats_sorted":["cs.DM","math.CO","physics.soc-ph"],"title_canon_sha256":"76bd9e5e1716ba25ae80d2d646c755b08cff20883bf3fc48b609ffa4a120f724","abstract_canon_sha256":"3a002c18840745dadf78cf6b97b009d915b98b30d6dc6bb8022abc6becbce0c4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:20:21.884650Z","signature_b64":"UL5kgczA8sgnBIQ3/QDEmM6p9yBA08gxJrQsFCyR9rtcvoen6pSIhwjHIB/yvTTt3L8P0pffLYCo7/5u0VxgCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a92c5d4cd54570e56c3111dcebfc450a599cc921647318a203912ac3ac6ac60c","last_reissued_at":"2026-05-18T04:20:21.884134Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:20:21.884134Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"High Degree Vertices, Eigenvalues and Diameter of Random Apollonian Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.CO","physics.soc-ph"],"primary_cat":"cs.SI","authors_text":"Alan Frieze, Charalampos E. 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