{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T06:02:14Z","timestamp":1771480934607,"version":"3.50.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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 $\\eta_r&gt;0$ such that $\\mathrm{diam}(D(n,r))=(1+\\eta_r+o(1))\\log_r{n}$. The constant $\\eta_r$ is related to branching processes and also appears in other models of random undirected graphs. 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 $\\pi_{\\min}=n^{-(1+\\eta_r)+o(1)}$. Our proof shows that the vertices with $\\pi(v)$ near to $\\pi_{\\min}$ lie at the top of \"narrow, slippery tower\"; such vertices are also responsible for increasing the diameter from $(1+o(1))\\log_r n$ to $(1+\\eta_r+o(1))\\log_r{n}$.<\/jats:p>","DOI":"10.37236\/9485","type":"journal-article","created":{"date-parts":[[2020,9,4]],"date-time":"2020-09-04T02:47:59Z","timestamp":1599187679000},"source":"Crossref","is-referenced-by-count":8,"title":["Diameter and Stationary Distribution of Random $r$-Out Digraphs"],"prefix":"10.37236","volume":"27","author":[{"given":"Louigi","family":"Addario-Berry","sequence":"first","affiliation":[]},{"given":"Borja","family":"Balle","sequence":"additional","affiliation":[]},{"given":"Guillem","family":"Perarnau","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2020,8,7]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i3p28\/8146","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i3p28\/8146","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,9,4]],"date-time":"2020-09-04T02:47:59Z","timestamp":1599187679000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i3p28"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,7]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,7,9]]}},"URL":"https:\/\/doi.org\/10.37236\/9485","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,8,7]]},"article-number":"P3.28"}}