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

Heat diffusion distance processes: a statistically founded method to analyze graph data sets 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Applied and Computational Topology Springer Journals

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

Lasalle, Etienne
Journal of Applied and Computational Topology , Volume OnlineFirst – May 18, 2023
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References (60)

Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
ISSN
2367-1726
eISSN
2367-1734
DOI
10.1007/s41468-023-00125-w
Publisher site
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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.

Journal

Journal of Applied and Computational Topology – Springer Journals

Published: May 18, 2023

Keywords: Graphs; Two-sample tests; Empirical processes; Topological data analysis; Primary 05C50; 60J60; 60F05; Secondary 55N31; 62G15; 62G10

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