On the length of the shortest path in a sparse Barak-Erdős graph

We consider an inhomogeneous version of the Barak-Erdős graph, i.e. a directed Erős-Rényi random graph on $\{1,\ldots,n\}$ with no loop. Given $f$ a Riemann-integrable non-negative function on $[0,1]^2$ and $γ> 0$, we define $G(n,f,γ)$ 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 $p_{i, j}^{(n)} = \frac{f(i/n,j/n)}{n^γ}$, 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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