The elliptic Harnack inequality for undirected graphs was proven by Delmotte in the paper "Inegalite de Harnack elliptique sur les graphes" (French, http://matwbn.icm.edu.pl/ksiazki/cm/cm72/cm7213.pdf). A recent generalization was given by Andres-Deuschel-Slowik (English, arXiv:1312.5473).
These results state that, under volume and isoperimetry conditions on an undirected weighted graph $G$, the maximum and minimum of a weighted-harmonic function $f$ on a ball $B(x, n)$ in $G$ satisfy
$$\max_{y \in B(x, n)} f(y) \leq C \min_{y \in B(x, n)} f(y)$$
for some constant $C$ and $B(x, n)$ being the points of graph distance at most $n$ for any $x \in G$.
I am interested in a generalization of this result (or any analogue of elliptic regularity) to directed graphs. In this case, the graph Laplacian should likely be replaced by the directed graph Laplacian defined by Chung (http://www.math.ucsd.edu/~fan/wp/dichee.pdf).
Is there a generalization of this type of result to the case of directed graphs?