{"paper":{"title":"Diameter and Stationary Distribution of Random $r$-out Digraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.CO"],"primary_cat":"math.PR","authors_text":"Borja Balle, Guillem Perarnau, Louigi Addario-Berry","submitted_at":"2015-04-26T15:58:22Z","abstract_excerpt":"Let $D(n,r)$ be a random $r$-out regular directed multigraph on the set of vertices $\\{1,\\ldots,n\\}$. In this work, we establish that for every $r \\ge 2$, there exists $\\eta_r>0$ such that $\\text{diam}(D(n,r))=(1+\\eta_r+o(1))\\log_r{n}$. Our techniques also allow us to bound some extremal quantities related to the stationary distribution of a simple random walk on $D(n,r)$. In particular, we determine the asymptotic behaviour of $\\pi_{\\max}$ and $\\pi_{\\min}$, the maximum and the minimum values of the stationary distribution. We show that with high probability $\\pi_{\\max} = n^{-1+o(1)}$ and $\\pi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.06840","kind":"arxiv","version":1},"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"}