Heat diffusion distance processes: a statistically founded method to analyze graph data sets

TL;DR

This paper introduces a statistically grounded multiscale method using heat diffusion to compare graph data sets without requiring node correspondence, enabling robust analysis of graphs of different sizes.

Contribution

It develops a novel heat diffusion-based framework for graph comparison, including statistical properties, confidence bands, and two-sample tests, applicable to graphs of varying sizes.

Findings

Proves a functional central limit theorem for the proposed processes.

Designs consistent bootstrap-based confidence bands and two-sample tests.

Demonstrates effectiveness through simulations on synthetic data.

Abstract

We propose two multiscale comparisons of graphs using heat diffusion, allowing to compare graphs without node correspondence or even with different sizes. These multiscale comparisons lead to the definition of Lipschitz-continuous empirical processes indexed by a real parameter. The statistical properties of empirical means of such processes are studied in the general case. Under mild assumptions, we prove a functional central limit theorem, as well as a Gaussian approximation with a rate depending only on the sample size. Once applied to our processes, these results allow to analyze data sets of pairs of graphs. We design consistent confidence bands around empirical means and consistent two-sample tests, using bootstrap methods. Their performances are evaluated by simulations on synthetic data sets.