Motivation: I'm reading this preprint, which takes a digraph $G = (V, E)$ and then builds a projective algebraic set $P(G)$ by assigning a variable to each edge and then defining certain polynomial "incidence relations" at each node that must be satisfied. The author then proves that the basic properties of $P(G)$ encode properties of cycle decompositions of $G$. For example, if $c$ is the maximum size of a set of edge-disjoint directed cycles in $G$, then $\dim P(G) = c-1$, and the degree of $P(G)$ tells you the number of different ways to achieve this maximum cycle decomposition.
Question: The author mentions that this argument is an analysis of the Khovanov-Rozansky Homology of $G$ (which is defined in the preprint at the bottom of p4). My background is in combinatorics rather than algebra and so I'm having trouble understanding the ways in which this argument is related to the existing theory of KR Homology. In particular, I have gone and understood the gist of the "original" KR Homology which is defined for links rather than graphs -- what is the relationship between this theory and the argument for graphs made in the above paper?