A Distance for Geometric Graphs via the Labeled Merge Tree Interleaving Distance

TL;DR

This paper introduces a new metric for measuring distances between geometric graphs using merge trees, incorporating a directional transform to preserve data features, with efficient computation methods demonstrated on real datasets.

Contribution

It proposes a novel distance measure for embedded graphs based on merge trees and introduces a directional transform to enhance information preservation.

Findings

The metric has desirable theoretical properties.

Approximate and exact computation methods are developed.

The approach is effective on real-world data sets.

Abstract

Geometric graphs appear in many real-world data sets, such as road networks, sensor networks, and molecules. We investigate the notion of distance between embedded graphs and present a metric to measure the distance between two geometric graphs via merge trees. In order to preserve as much useful information as possible from the original data, we introduce a way of rotating the sublevel set to obtain the merge trees via the idea of the directional transform. We represent the merge trees using a surjective multi-labeling scheme and then compute the distance between two representative matrices. We show some theoretically desirable qualities and present two methods of computation: approximation via sampling and exact distance using a kinetic data structure, both in polynomial time. We illustrate its utility by implementing it on two data sets.