The Walk Distances in Graphs

TL;DR

This paper explores walk distances in graphs, showing their relation to shortest path and resistance distances, and introduces their properties and subclasses, enhancing understanding of graph proximity measures.

Contribution

It establishes the connection between walk distances, logarithmic forest distances, and resistance distances, and analyzes their convergence and properties.

Findings

Walk distances are graph-geodetic and converge to shortest path and long walk distances.

Logarithmic forest distances are a subclass of walk distances.

Long walk distance equals resistance distance in a transformed graph.

Abstract

The walk distances in graphs are defined as the result of appropriate transformations of the $\sum_{k=0}^\infty(tA)^k$ proximity measures, where $A$ is the weighted adjacency matrix of a graph and $t$ is a sufficiently small positive parameter. The walk distances are graph-geodetic; moreover, they converge to the shortest path distance and to the so-called long walk distance as the parameter $t$ approaches its limiting values. We also show that the logarithmic forest distances which are known to generalize the resistance distance and the shortest path distance are a subclass of walk distances. On the other hand, the long walk distance is equal to the resistance distance in a transformed graph.