A Weighted Quiver Kernel using Functor Homology

TL;DR

This paper introduces a novel homological approach to analyze weighted directed networks, enabling the definition of a new graph kernel and demonstrating practical applicability through real data experiments.

Contribution

It presents a new homological method for weighted directed networks and a graph kernel based on this approach, with practical computations on real data.

Findings

Method is practically applicable to real data

Homology groups vanish for acyclic graphs in high dimensions

New graph kernel effectively captures weighted directed network features

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.