A Weighted Quiver Kernel using Functor Homology
This work addresses the analysis of weighted directed networks, offering a novel mathematical approach that could benefit researchers in network theory, but it appears incremental as it builds on known homological algebra methods.
The paper tackles the problem of analyzing weighted directed networks by proposing a homological method to define homology groups for such graphs, which leads to a new graph kernel; sample computations with real data show practical applicability.
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.