Let $G$ be a finite directed graph with weighted edges and an initial vertex $v_0$. Assume that there exists at least one walk of length $n$ starting at $v_0$ for each $n \ge 1$. (A walk is a sequence of vertices such that each vertex is adjacent to the next vertex in the sequence. It can visit the same vertex more than once.)
For each $n \ge 1$, let $f(n)$ denote the maximum weight of a walk of length $n$, starting at $v_0$. I conjecture that there exist positive integers $N$, $p$, and $c$ such that $f(n+p) - f(n) = c$ for all $n \ge N$.
Can anyone provide a proof of this conjecture, or (preferably) a reference that contains such a proof?