On topological descriptors for graph products

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.

💡 Research Summary

This paper investigates the use of topological descriptors—Euler characteristic (EC) and persistent homology (PH)—on graph products, focusing on the Cartesian (box) product of two colored graphs. The authors first formalize color‑based filtrations at both the vertex and edge levels, defining functions that assign real values to vertices (via a vertex‑color map) and to edges (either as the maximum of incident vertex values or via a symmetric edge‑color map). These filtrations induce a sequence of subgraphs whose topological features can be tracked.

The core theoretical contribution concerns the expressive power of EC. By analyzing the combinatorial structure of colored graphs, the authors prove that two graphs have identical EC diagrams for all possible color functions if and only if they have identical counts of vertices of each color and identical counts of edges for each unordered pair of colors. This result is encapsulated in Theorem 2 and its corollaries. Moreover, they introduce a “max EC” variant that restricts edge filtrations to the maximum of incident vertex values; Proposition 1 and Theorem 1 show that max EC has exactly the same discriminative ability as full EC, while being computationally cheaper. The same equivalence extends to colored simplicial complexes (Theorem 3).

In contrast, PH exhibits strictly greater expressive power on graph products. The authors construct explicit examples of non‑isomorphic graphs G and H that cannot be distinguished by PH on the individual graphs, yet their self‑products G □ G and H □ H generate different 1‑dimensional persistence diagrams (Proposition 4). This demonstrates that the product operation can create new cycles (or eliminate them) that are invisible in the components, thereby enriching the PH signature. Conversely, EC does not gain any extra discriminative information from the product (Proposition 5).

A substantial part of the paper is devoted to algorithmic development. The product of vertex filtrations is shown to correspond to taking the pointwise maximum of the two vertex‑color functions, while the product of edge filtrations corresponds to a specific edge‑coloring on the product graph (Propositions 6 and 7). Leveraging these observations, the authors design linear‑time algorithms for extracting 0‑th persistence pairs (Theorem 4) and 1‑st persistence pairs (Theorem 5) from product filtrations. The algorithms only need to process the finite set of critical filtration values, leading to significant speed‑ups compared with naïve PH computation on the full product graph.

Empirical evaluation validates the theoretical claims. Runtime experiments confirm that the proposed product‑PH algorithms are orders of magnitude faster than generic PH libraries on large graphs. Expressivity tests on synthetic datasets illustrate that PH on products can separate graph classes that are indistinguishable by EC or by PH on the factors alone. Finally, the authors integrate EC, max EC, and product‑PH features into graph neural networks for standard benchmark classification tasks (e.g., MUTAG, PROTEINS). The inclusion of product‑PH descriptors yields consistent accuracy improvements, especially on datasets where relational structures naturally decompose into product forms.

Overall, the paper makes three key contributions: (1) a complete combinatorial characterization of EC’s expressive power and its equivalence to max EC; (2) a proof that PH on graph products strictly surpasses the information content of PH on individual graphs; and (3) efficient algorithms for computing PH on vertex‑ and edge‑level product filtrations, backed by extensive experiments. These results open a pathway for incorporating powerful, yet computationally tractable, topological features into graph learning pipelines, particularly when the underlying data exhibit product‑like structure. Future work may explore learnable color functions, extensions to other graph products, and integration with higher‑order message‑passing frameworks.