Diameter and stationary distribution of random r-out digraphs
Let D(n, r) be a random r-out regular directed multigraph on the set of vertices {1, . . . , n}. In this work, we establish that for every r = 2, there exists ¿r > 0 such that diam(D(n, r)) = (1 + ¿r + o(1)) logr n. The constant ¿r is related to branching processes and also appears in other model...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2020 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/342134 |
| Acceso en línea: | https://hdl.handle.net/2117/342134 https://dx.doi.org/10.37236/9485 |
| Access Level: | acceso abierto |
| Palabra clave: | Graph theory Grafs, Teoria de Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs |
| Sumario: | Let D(n, r) be a random r-out regular directed multigraph on the set of vertices {1, . . . , n}. In this work, we establish that for every r = 2, there exists ¿r > 0 such that diam(D(n, r)) = (1 + ¿r + o(1)) logr n. The constant ¿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 pmax and pmin, the maximum and the minimum values of the stationary distribution. We show that with high probability pmax = n -1+o(1) and pmin = n -(1+¿r)+o(1). Our proof shows that the vertices with p(v) near to pmin lie at the top of “narrow, slippery towers”; such vertices are also responsible for increasing the diameter from (1+o(1)) logr n to (1 + ¿r + o(1)) logr n. |
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