On topological descriptors for graph products

TL;DR

The work studies how topological descriptors—persistent homology (PH) and Euler characteristic (EC)—behave under graph box products with color-based filtrations. It proves that EC has the same expressive power under max-EC as EC and shows that PH on product graphs can capture information beyond what the components provide, while EC does not, accompanied by efficient PH computation algorithms for vertex- and edge-based product filtrations. It develops a formal theory of product filtrations, including vertex- and edge-level constructions, and demonstrates practical benefits in runtime and graph classification when using product-based topological descriptors in GNN pipelines. Overall, the paper provides a principled calculus for enriching graph representations with product-filtered topological descriptors, with broad implications for isomorphism testing, relational data analysis, and graph learning.

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 descriptors - 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 products contain strictly more information than the computation on individual graphs, whereas EC does not. Additionally, we provide algorithms to compute the PH diagrams 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.

Figures (5)

• Figure 1: Graphs $G$ and $H$ that vertex-level and edge-level PH cannot tell apart, but they can be differentiated by running edge-level PH on $G \Box G$ and $H \Box H$. Here, connecting the edges between vertices of color $(\operatorname{blue}, \operatorname{blue})$ and $(\operatorname{red}, \operatorname{blue})$ creates a non-trivial cycle in $(A)$ and no cycles in $(B)$.
• Figure 2: The product of the vertex filtrations on $G = H$ given by $f_v(\operatorname{red}) > f_v(\operatorname{blue})$. See Example \ref{['exp::vertex_PH_algo_example']} for how the algorithm in Theorem \ref{['thm::vertex_PH_prod']} is used to compute the PH of this filtration.
• Figure 3: Average runtime (in sec.) of different implementations of topological descriptors for graph products. Our algorithms (Theorems 4 and 5) consistently outperform previous implementations.
• Figure 4: The product of the edge filtrations on $G = H$ given by $f_e(\operatorname{red}, \operatorname{red}) = 2 > f_e(\operatorname{red},\operatorname{blue}) = 1$. See Example \ref{['exp::edge_PH_algo_example']} for how the algorithm in Theorem \ref{['thm::betti_number_prod']} is used to compute the PH of this filtration.
• Figure 5: Graph $G$ in Example \ref{['exp::ec']}.

Theorems & Definitions (52)

• Definition 1: Color-Based Filtrations
• Definition 2
• Definition 3
• Definition 4
• Proposition 1
• Theorem 1
• Proposition 2: Characterization of (max) EC Edge Filtrations
• Proposition 3: Characterization of (max) EC Vertex Filtrations
• Theorem 2: Characterization of (max) EC Diagrams
• Theorem 3