Theoretical analysis and computation of the sample Frechet mean for sets of large graphs based on spectral information

TL;DR

This paper introduces a spectral-based pseudometric for sets of graphs, enabling the computation of the Frechet mean to analyze centrality in graph data, with an efficient algorithm for large graphs.

Contribution

It proposes a novel spectral pseudometric for graph sets and develops an algorithm to approximate the Frechet mean using this metric.

Findings

The spectral pseudometric captures multi-scale structural changes.

An efficient algorithm for computing the Frechet mean of large graphs is presented.

The method is suitable for statistical analysis of graph-valued data.

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 Frechet mean. In this work, we equip a set of graphs with the pseudometric defined by the 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 for graph-valued data. We describe an algorithm to compute an approximation to the sample Frechet mean of a set of undirected unweighted graphs with a fixed size using this pseudometric.