On Proximity Measures for Graph Vertices

TL;DR

This paper analyzes various proximity measures for graph vertices, extending classical distances to weighted and directed graphs, and provides characterizations and interpretations of these measures.

Contribution

It introduces and characterizes several proximity measures for weighted multigraphs and multidigraphs, including their topological interpretation via the Laplacian matrix inverse.

Findings

Proximity measures account for all connections, not just shortest paths.

A topological interpretation of the generalized inverse of the Laplacian matrix is provided.

Conditions characterizing different proximity measures are established.

Abstract

We study the properties of several proximity measures for the vertices of weighted multigraphs and multidigraphs. Unlike the classical distance for the vertices of connected graphs, these proximity measures are applicable to weighted structures and take into account not only the shortest, but also all other connections, which is desirable in many applications. To apply these proximity measures to unweighted structures, every edge should be assigned the same weight which determines the proportion of taking account of two routes, from which one is one edge longer than the other. Among the proximity measures we consider path accessibility, route accessibility, relative forest accessibility along with its components, accessibility via dense forests, and connection reliability. A number of characteristic conditions is introduced and employed to characterize the proximity measures. A topological interpretation is obtained for the Moore-Penrose generalized inverse of the Laplacian matrix of a weighted multigraph.