Matrices of Forests and the Analysis of Digraphs

TL;DR

This paper explores matrices derived from spanning rooted forests to analyze digraph structures, revealing properties like vertex proximity and ranking, with applications in information dissemination and preference analysis.

Contribution

It introduces a novel approach using spanning forest matrices for digraph analysis, including new measures of vertex accessibility and interpretations for information flow.

Findings

Vertex accessibility measure has desirable properties

Normalized out-forest matrix interprets information dissemination

Method effectively reveals bicomponents and rankings

Abstract

The matrices of spanning rooted forests are studied as a tool for analysing the structure of digraphs and measuring their characteristics. The problems of revealing the basis bicomponents, measuring vertex proximity, and ranking from preference relations / sports competitions are considered. It is shown that the vertex accessibility measure based on spanning forests has a number of desirable properties. An interpretation for the normalized matrix of out-forests in terms of information dissemination is given. Keywords: Laplacian matrix, spanning forest, matrix-forest theorem, proximity measure, bicomponent, ranking, incomplete tournament, paired comparisons