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

TL;DR

This paper investigates the asymptotic behavior of the shortest path length in an inhomogeneous, sparse directed random graph model based on a scaled probability function, extending classical results to more general settings.

Contribution

It introduces a new inhomogeneous sparse graph model with a general kernel function and analyzes the asymptotic shortest path length, broadening understanding beyond homogeneous cases.

Findings

Asymptotic behavior of shortest path length characterized

Dependence on the kernel function and sparsity parameter established

Results extend classical Erdős-Rényi shortest path results

Abstract

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 $n \to \infty$.