Suppose $G=\langle V,E \rangle$ is a directed graph consisting of $n\in \mathbb{N}$ vertices. Vertex $v_i \in V$ has an edge to vertex $v_j \in V$ with a probability of $P(i, j) = f(|i-j|)$ where $f$ is a descending "good enough" function.
What is probability to have a path of length $k$?
Update 1. Let us consider a simplified problem. Suppose the probability of having an edge between any two arbitrary vertices is fixed and equals $p$. Thus, the adjacency matrix $\bf A$ consists of elements $a_{ij} = \delta(p)$ equal $1$ with a probability of $p$.
It's well known that $({\bf A}^k)_{ij}$ gives the number of $k$-paths from $v_i$ to $v_j$. Can we employ that?