Graph Pseudometrics from a Topological Point of View

TL;DR

This paper investigates pseudometrics for directed graphs to efficiently approximate their topological features, such as Betti numbers, facilitating analysis of complex networks without costly computations.

Contribution

It introduces methods to evaluate how effectively graph pseudometrics capture the topology of directed graphs, enabling more efficient topological analysis.

Findings

Certain pseudometrics accurately reflect topological differences

Evaluation methods work across various random graph families

Some pseudometrics are computationally efficient for topology detection

Abstract

We explore pseudometrics for directed graphs in order to better understand their topological properties. The directed flag complex associated to a directed graph provides a useful bridge between network science and topology. Indeed, it has often been observed that phenomena exhibited by real-world networks reflect the topology of their flag complexes, as measured, for example, by Betti numbers or simplex counts. As it is often computationally expensive (or even unfeasible) to determine such topological features exactly, it would be extremely valuable to have pseudometrics on the set of directed graphs that can both detect the topological differences and be computed efficiently. To facilitate work in this direction, we introduce methods to measure how well a graph pseudometric captures the topology of a directed graph. We then use these methods to evaluate some well-established pseudometrics, using test data drawn from several families of random graphs.