A Weighted Quiver Kernel using Functor Homology

Abstract: In this paper, we propose a new homological method to study weighted directed networks. Our model of such networks is a directed graph $Q$ equipped with a weight function $w$ on the set $Q_{1}$ of arrows in $Q$. We require that the range $W$ of our weight function is equipped with an addition or a multiplication, i.e., $W$ is a monoid in the mathematical terminology. When $W$ is equipped with a representation on a vector space $M$, the standard method of homological algebra allows us to define the homology groups $H_{*}(Q,w;M)$. It is known that when $Q$ has no oriented cycles, $H_{n}(Q,w;M)=0$ for $n\ge 2$ and $H_{1}(Q,w;M)$ can be easily computed. This fact allows us to define a new graph kernel for weighted directed graphs. We made two sample computations with real data and found that our method is practically applicable.