The graph bottleneck identity

TL;DR

This paper introduces a class of graph-geodetic distances derived from transitional measures on directed graphs, unifying shortest-path and resistance distances through a logarithmic transformation of matrix-based measures.

Contribution

It defines the graph bottleneck identity and demonstrates how positive transitional measures produce a new family of distances with desirable properties, extending existing graph distance concepts.

Findings

Every positive transitional measure yields a graph-geodetic distance.

The approach unifies shortest-path and resistance distances.

Multiple matrix types determine transitional measures for digraphs.

Abstract

A matrix $S=(s_{ij})\in{\mathbb R}^{n\times n}$ is said to determine a \emph{transitional measure} for a digraph $G$ on $n$ vertices if for all $i,j,k\in\{1,\...,n\},$ the \emph{transition inequality} $s_{ij} s_{jk}\le s_{ik} s_{jj}$ holds and reduces to the equality (called the \emph{graph bottleneck identity}) if and only if every path in $G$ from $i$ to $k$ contains $j$. We show that every positive transitional measure produces a distance by means of a logarithmic transformation. Moreover, the resulting distance $d(\cdot,\cdot)$ is \emph{graph-geodetic}, that is, $d(i,j)+d(j,k)=d(i,k)$ holds if and only if every path in $G$ connecting $i$ and $k$ contains $j$. Five types of matrices that determine transitional measures for a digraph are considered, namely, the matrices of path weights, connection reliabilities, route weights, and the weights of in-forests and out-forests. The results obtained have undirected counterparts. In [P. Chebotarev, A class of graph-geodetic distances generalizing the shortest-path and the resistance distances, Discrete Appl. Math., URL http://dx.doi.org/10.1016/j.dam.2010.11.017] the present approach is used to fill the gap between the shortest path distance and the resistance distance.