Comparing Embedded Graphs Using Average Branching Distance

TL;DR

This paper introduces the average branching distance, a new method for comparing embedded graphs by analyzing their structure across multiple directions, and demonstrates its practical effectiveness despite some theoretical limitations.

Contribution

It proposes the average branching distance, a novel approach for graph comparison that extends previous merge tree distances by averaging over many directions.

Findings

The new distance is useful for clustering embedded graphs.

Theoretical issues with the distance do not hinder practical applications.

Open-source code is provided for implementation.

Abstract

Graphs drawn in the plane are ubiquitous, arising from data sets through a variety of methods ranging from GIS analysis to image classification to shape analysis. A fundamental problem in this type of data is comparison: given a set of such graphs, can we rank how similar they are, in such a way that we capture their geometric "shape" in the plane? In this paper we explore a method to compare two such embedded graphs, via a simplified combinatorial representation called a tail-less merge tree which encodes the structure based on a fixed direction. First, we examine the properties of a distance designed to compare merge trees called the branching distance, and show that the distance as defined in previous work fails to satisfy some of the requirements of a metric. We incorporate this into a new distance function called average branching distance to compare graphs by looking at the branching distance for merge trees defined over many directions. Despite the theoretical issues, we show that the definition is still quite useful in practice by using our open-source code to cluster data sets of embedded graphs.