Approximate Fr\'echet Mean for Data Sets of Sparse Graphs

TL;DR

This paper introduces a spectral pseudometric for graphs based on eigenvalues and proposes an algorithm to approximate the Fréchet mean of a set of graphs, enabling better statistical analysis of graph data.

Contribution

It presents a novel spectral pseudometric for graphs and an algorithm to approximate the Fréchet mean, facilitating statistical analysis of graph sets.

Findings

Spectral pseudometric captures multi-scale structural changes.

Algorithm efficiently approximates the Fréchet mean for fixed-size graphs.

Enables new statistical tools for graph data analysis.

Abstract

To characterize the location (mean, median) of a set of graphs, one needs a notion of centrality that is adapted to metric spaces, since graph sets are not Euclidean spaces. A standard approach is to consider the Fr\'echet mean. In this work, we equip a set of graph with the pseudometric defined by the $\ell_2$ norm between the eigenvalues of their respective adjacency matrix . Unlike the edit distance, this pseudometric reveals structural changes at multiple scales, and is well adapted to studying various statistical problems on sets of graphs. We describe an algorithm to compute an approximation to the Fr\'echet mean of a set of undirected unweighted graphs with a fixed size.