Modulus of time-respecting paths

TL;DR

This paper introduces a temporal version of p-modulus for time-respecting paths in graphs, incorporating time penalties to reflect traversal duration and enabling analysis of temporal network dynamics.

Contribution

It extends the p-modulus concept to temporal graphs, linking it to static graph problems and allowing the use of existing theories for temporal path analysis.

Findings

Temporal p-modulus captures both path length and diversity.

Transformation links temporal modulus to static graph problems.

Properties demonstrated on example temporal graphs.

Abstract

On a static graph, the p-modulus of a family of paths reflects both the lengths of these paths as well as their diversity; a family of many short, disjoint paths has larger modulus than a family of a few long overlapping paths. In this work, we define a version of p-modulus for time-respecting paths on temporal graphs. This formulation makes use of a time penalty function as a means of discounting paths that take a relatively long time to traverse, thus allowing modulus to capture temporal information about the family as well. By means of a transformation, we show that this temporal p-modulus can be recognized as a p-modulus problem on a static graph and, therefore, that much of the known theory of p-modulus of families of objects can be translated to the case of temporal paths. We demonstrate some properties of temporal modulus on examples.