The Matrix of Maximum Out Forests of a Digraph and Its Applications

TL;DR

This paper introduces the concept of maximum out forests in weighted digraphs, explores their matrix properties, and demonstrates their applications in Markov chains, preference aggregation, and graph structure analysis.

Contribution

It provides a new proof of the Markov chain tree theorem and links maximum out forests to various practical problems in graph theory and Markov processes.

Findings

Matrix of maximum out forests equals Cesàro limiting probabilities in Markov chains

Applications in preference aggregation and vertex proximity measurement

New insights into digraph structure analysis

Abstract

We study the maximum out forests of a (weighted) digraph and the matrix of maximum out forests. A maximum out forest of a digraph G is a spanning subgraph of G that consists of disjoint diverging trees and has the maximum possible number of arcs. If a digraph contains any out arborescences, then maximum out forests coincide with them. We provide a new proof to the Markov chain tree theorem saying that the matrix of Ces`aro limiting probabilities of an arbitrary stationary finite Markov chain coincides with the normalized matrix of maximum out forests of the weighted digraph that corresponds to the Markov chain. We discuss the applications of the matrix of maximum out forests and its transposition, the matrix of limiting accessibilities of a digraph, to the problems of preference aggregation, measuring the vertex proximity, and uncovering the structure of a digraph.