A generalized inverse for graphs with absorption

TL;DR

This paper introduces the absorption inverse, a new generalized inverse of the graph Laplacian for weighted, directed graphs with absorption, linking graph structure, absorption rates, and applications like metrics and centrality.

Contribution

It defines the absorption inverse, relates it to spanning forests and absorbing random walks, and demonstrates its applications in graph analysis.

Findings

The absorption inverse captures both graph structure and absorption rates.

It relates to the fundamental matrix of absorbing random walks.

Applications include a directed distance metric, spectral partitioning, and centrality measures.

Abstract

We consider weighted, directed graphs with a notion of absorption on the vertices, related to absorbing random walks on graphs. We define a generalized inverse of the graph Laplacian, called the absorption inverse, that reflects both the graph structure as well as the absorption rates on the vertices. Properties of this generalized inverse are presented, including a matrix forest theorem relating this generalized inverse to spanning forests of a related graph, as well as relationships between the absorption inverse and the fundamental matrix of the absorbing random walk. Applications of the absorption inverse for describing the structure of graphs with absorption are presented, including a directed distance metric, spectral partitioning algorithm, and centrality measure.