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Article Dans Une Revue Statistics and Probability Letters Année : 2022

On the length of the shortest path in a sparse Barak-Erd\H{o}s graph

Résumé

We consider an inhomogeneous version of the Barak-Erd\H{o}s graph, i.e. a directed Er\H{o}s-R\'enyi random graph on $\{1,\ldots,n\}$ with no loop. Given $f$ a Riemann-integrable non-negative function on $[0,1]^2$ and $\gamma > 0$, we define $G(n,f,\gamma)$ as the random graph with vertex set $\{1,\ldots,n\}$ such that for each $i < j$ the directed edge $(i,j)$ is present with probability $ p_{i, j}^{(n)} = \frac{f(i/n,j/n)}{n^\gamma}$, independently of any other edge. We denote by $L_n$ the length of the shortest path between vertices $1$ and $n$, and take interest in the asymptotic behaviour of $L_n$ as $n \to \infty$.
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Dates et versions

hal-03514819 , version 1 (06-01-2022)
hal-03514819 , version 2 (17-11-2022)

Identifiants

Citer

Bastien Mallein, Pavel Tesemnikov. On the length of the shortest path in a sparse Barak-Erd\H{o}s graph. Statistics and Probability Letters, 2022, 190, pp.ID 109634. ⟨hal-03514819v2⟩
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