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We consider an inhomogeneous version of the Barak-Erd\H{o}s graph, i.e. adirected 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 withprobability $ p_{i, j}^{(n)} = \frac{f(i/n,j/n)}{n^\gamma}$, independently ofany other edge. We denote by $L_n$ the length of the shortest path betweenvertices $1$ and $n$, and take interest in the asymptotic behaviour of $L_n$ as$n \to \infty$.

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