{"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. Tsourakakis","submitted_at":"2011-04-27T21:55:21Z","abstract_excerpt":"In this work we analyze basic properties of Random Apollonian Networks \\cite{zhang,zhou}, a popular stochastic model which generates planar graphs with power law properties. 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 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.5259","kind":"arxiv","version":3},"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"}