{"paper":{"title":"The Asymptotics of Large Constrained Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.SI","math-ph","math.MP"],"primary_cat":"math.CO","authors_text":"Charles Radin, Kui Ren, Lorenzo Sadun","submitted_at":"2014-01-06T19:11:41Z","abstract_excerpt":"We show, through local estimates and simulation, that if one constrains simple graphs by their densities $\\varepsilon$ of edges and $\\tau$ of triangles, then asymptotically (in the number of vertices) for over $95\\%$ of the possible range of those densities there is a well-defined typical graph, and it has a very simple structure: the vertices are decomposed into two subsets $V_1$ and $V_2$ of fixed relative size $c$ and $1-c$, and there are well-defined probabilities of edges, $g_{jk}$, between $v_j\\in V_j$, and $v_k\\in V_k$. Furthermore the four parameters $c, g_{11}, g_{22}$ and $g_{12}$ ar"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.1170","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"}