# A linear-time algorithm and analysis of graph Relative Hausdorff distance

## TL;DR

This paper introduces the first linear-time algorithm for computing the graph Relative Hausdorff distance, analyzes its properties, and demonstrates its effectiveness in comparing real-world and structured networks.

## Contribution

It provides a novel linear-time algorithm for RH distance and offers a comprehensive analysis of its properties and behavior.

## Key findings

- Linear-time algorithm for RH distance computation
- Analysis of RH distance properties and extremal behavior
- Empirical evaluation on real-world and structured graphs

## Abstract

Graph similarity metrics serve far-ranging purposes across many domains in data science. As graph datasets grow in size, scientists need comparative tools that capture meaningful differences, yet are lightweight and scalable. Graph Relative Hausdorff (RH) distance is a promising, recently proposed measure for quantifying degree distribution similarity. In spite of recent interest in RH distance, little is known about its properties. Here, we conduct an algorithmic and analytic study of RH distance. In particular, we provide the first linear-time algorithm for computing RH distance, analyze examples of RH distance between pairs of real-world networks as well as structured families of graphs, and prove several analytic results concerning the range, density, and extremal behavior of RH distance values.