Assignment Based Metrics for Attributed Graphs

TL;DR

This paper introduces new metrics, GTT and GOSPA, for comparing attributed graphs of different sizes, capturing both structure and attributes, with applications in statistical testing of biological data.

Contribution

The paper presents novel assignment-based metrics for attributed graphs, including theoretical properties, computational algorithms, and an application to biological data analysis.

Findings

GOSPA effectively detects differences in biological graphs.

Algorithms for computing GOSPA are efficient and heuristic methods improve performance.

The metrics relate well to existing graph distance measures.

Abstract

We introduce the Graph TT (GTT) and Graph OSPA (GOSPA) metrics based on optimal assignment, which allow us to compare not only the edge structures but also general vertex and edge attributes of graphs of possibly different sizes. We argue that this provides an intuitive and universal way to measure the distance between finite simple attributed graphs. Our paper discusses useful equivalences and inequalities as well as the relation of the new metrics to various existing quantifications of distance between graphs. By deriving a representation of a graph as a pair of point processes, we are able to formulate and study a new type of (finite) random graph convergence and demonstrate its applicability using general point processes of vertices with independent random edges. Computational aspects of the new metrics are studied in the form of an exact and two heuristic algorithms that are derived from previous algorithms for similar tasks. As an application, we perform a statistical test based on the GOSPA metric for functional differences in olfactory neurons of Drosophila flies.