NeurIPS 2025

On topological descriptors for graph products
Mattie Ji
University of Pennsylvania
mji13@sas.upenn.edu
Amauri H. Souza
Federal Institute of Ceará
amauriholanda@ifce.edu.br
Vikas Garg
Aalto University
YaiYai Ltd
vgarg@csail.mit.edu
Abstract
Topological descriptors have been increasingly utilized for capturing multiscale
structural information in relational data. In this work, we consider various filtrations
on the (box) product of graphs and the effect on their outputs on the topological de-
scriptors - the Euler characteristic (EC) and persistent homology (PH). In particular,
we establish a complete characterization of the expressive power of EC on general
color-based filtrations. We also show that the PH descriptors of (virtual) graph prod-
ucts contain strictly more information than the computation on individual graphs,
whereas EC does not. Additionally, we provide algorithms to compute the PH dia-
grams of the product of vertex- and edge-level filtrations on the graph product. We
also substantiate our theoretical analysis with empirical investigations on runtime
analysis, expressivity, and graph classification performance. Overall, this work
paves way for powerful graph persistent descriptors via product filtrations. Code is
available at https://github.com/Aalto-QuML/tda_graph_product.
1
Introduction
Message-passing GNNs are prominent models for graph representation learning [19]. However, they
are bounded in expressivity by the WL hierarchy [33–35, 50] and cannot compute fundamental graph
properties such as cycles or connected components [10, 18]. Topological descriptors (TDs) such
as those based on persistent homology (PH) and the Euler characteristics (EC) can provide such
information and thus are being increasingly employed to augment GNNs, boosting their empirical
performance [7–9, 24, 48, 51].
In this work, we study two prominent types of topological descriptors based on PH [1, 13] and EC
[40, 42, 46] respectively. Both descriptors are defined with respect ...