We consider an inhomogeneous version of the Barak–Erdős graph, i.e. a directed Erdős–Rényi random graph on $\{1,\dots ,n\}$ 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 $\{1,\dots ,n\}$ such that for each $i<j$ the directed edge $(i,j)$ is present with probability $p_{i,j}^{\left(n\right)}=\frac{f(i/n,j/n)}{n^{\gamma }}$, 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 $n\to \infty$.

中文翻译：

---

关于稀疏 Barak-Erdős 图中最短路径的长度

我们考虑 Barak-Erdős 图的非齐次版本，即有向 Erdős-Rényi 随机图$\{1,\dots ,n\}$没有循环。给定$F$一个黎曼可积非负函数${[0,1]}^2$和$\gamma >0$，我们定义$G(n,F,\gamma )$作为具有顶点集的随机图$\{1,\dots ,n\}$这样对于每个$\mathrm{一世}<j$有向边$(\mathrm{一世},j)$有可能存在$p_{\mathrm{一世},j}^{\left(n\right)}=\frac{F(\mathrm{一世}/n,j/n)}{n^{\gamma }}$，独立于任何其他边。我们表示${\mathrm{大号}}_n$顶点 1 和顶点之间的最短路径的长度$n$，并对 的渐近行为感兴趣${\mathrm{大号}}_n$作为$n\to \infty$.