Geodesic
On the asymptotics for the minimal distance between extreme vertices in a generalised Barak--Erd\"{o}s graph
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 15 (2018), pp. 1556-1565

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We consider a generalization of the Barak–Erdös random graph, which is a graph with an ordered set of vertices $ \{ 0, 1, \ldots n \} $ and with directed edges from $ i $ to $ j $ for $ i j $ only, where each edge is present with a given probability $ p \in (0, 1) $. In our setting, probabilities $ p=p_{i,j} $ depend on distances $ j - i $ and may tend to $ 0 $ as $ j - i \to \infty $. We study the asymptotics for the distribution of the minimal path length between $ 0 $ and $ n $, when $ n $ becomes large.
Keywords: random graph, Barak–Erdös directed graph, minimal distance, boundary points, graph connectivity, first-passage percolation. 
@article{SEMR_2018_15_a41,
     author = {P. I. Tesemnikov},
     title = {On the asymptotics for the minimal distance between extreme vertices in a generalised {Barak--Erd\"{o}s} graph},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1556--1565},
     publisher = {mathdoc},
     volume = {15},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2018_15_a41/}
}
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P. I. Tesemnikov. On the asymptotics for the minimal distance between extreme vertices in a generalised Barak--Erd\"{o}s graph. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 15 (2018), pp. 1556-1565. http://geodesic.mathdoc.fr/item/SEMR_2018_15_a41/