{"paper":{"title":"Multipodal Structure and Phase Transitions in Large Constrained Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.SI","math-ph","math.MP","math.PR"],"primary_cat":"math.CO","authors_text":"Charles Radin, Kui Ren, Lorenzo Sadun, Richard Kenyon","submitted_at":"2014-05-03T15:06:49Z","abstract_excerpt":"We study the asymptotics of large, simple, labeled graphs constrained by the densities of edges and of $k$-star subgraphs, $k\\ge 2$ fixed. We prove that under such constraints graphs are \"multipodal\": asymptotically in the number of vertices there is a partition of the vertices into $M < \\infty$ subsets $V_1, V_2, \\ldots, V_M$, and a set of well-defined probabilities $g_{ij}$ of an edge between any $v_i \\in V_i$ and $v_j \\in V_j$. For $2\\le k\\le 30$ we determine the phase space: the combinations of edge and $k$-star densities achievable asymptotically. For these models there are special points"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.0599","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"}