Metric spaces of walks and Lipschitz duality on graphs

TL;DR

This paper develops a metric framework for analyzing walks on graphs as Lipschitz sequences, enabling the extension of Lipschitz functions and applications in proximity estimation and reinforcement learning.

Contribution

It introduces a weighted metric for walks on graphs, analyzes their properties, and provides tools for extending Lipschitz functions and estimating proximities in network structures.

Findings

Defined a weighted metric for walks on graphs

Provided representation formulas for proximities

Enabled Lipschitz function extension on walk spaces

Abstract

We study the metric structure of walks on graphs, understood as Lipschitz sequences. To this end, a weighted metric is introduced to handle sequences, enabling the definition of distances between walks based on stepwise vertex distances and weighted norms. We analyze the main properties of these metric spaces, which provides the foundation for the analysis of weaker forms of instruments to measure relative distances between walks: proximities. We provide some representation formulas for such proximities under different assumptions and provide explicit constructions for these cases. The resulting metric framework allows the use of classical tools from metric modeling, such as the extension of Lipschitz functions from subspaces of walks, which permits extending proximity functions while preserving fundamental properties via the mentioned representations. Potential applications include the estimation of proximities and the development of reinforcement learning strategies based on exploratory walks, offering a robust approach to Lipschitz regression on network structures.