Tractable $n$-Metrics for Multiple Graphs

TL;DR

This paper introduces a new family of multi-distances for multiple graphs that satisfy generalized metric properties and alignment consistency, enabling scalable and theoretically sound graph comparison methods.

Contribution

It presents the first multi-distances for graphs that combine alignment consistency with generalized metric properties, and shows they can be optimized via convex relaxation.

Findings

Existence of multi-distances satisfying generalized metric and alignment properties.

Multi-distances can be formulated as convex optimization problems.

Potential for improved graph comparison in machine learning applications.

Abstract

Graphs are used in almost every scientific discipline to express relations among a set of objects. Algorithms that compare graphs, and output a closeness score, or a correspondence among their nodes, are thus extremely important. Despite the large amount of work done, many of the scalable algorithms to compare graphs do not produce closeness scores that satisfy the intuitive properties of metrics. This is problematic since non-metrics are known to degrade the performance of algorithms such as distance-based clustering of graphs (Stratis and Bento 2018). On the other hand, the use of metrics increases the performance of several machine learning tasks (Indyk et al. 1999, Clarkson et al. 1999, Angiulli et al. 2002, Ackermann et al. 2010). In this paper, we introduce a new family of multi-distances (a distance between more than two elements) that satisfies a generalization of the properties of metrics to multiple elements. In the context of comparing graphs, we are the first to show the existence of multi-distances that simultaneously incorporate the useful property of alignment consistency (Nguyen et al. 2011), and a generalized metric property. Furthermore, we show that these multi-distances can be relaxed to convex optimization problems, without losing the generalized metric property.