Line Graph Vietoris-Rips Persistence Diagram for Topological Graph Representation Learning

TL;DR

This work addresses the gap where standard GNNs capture local node information but miss global graph topology. It introduces the Topological Edge Diagram (TED), an edge-filtration–based persistence diagram, and its neural realization LGVR (Line Graph Vietoris–Rips Persistence Diagram) to preserve node embeddings while encoding topology. The authors present two model frameworks, -LGVR and -LVGR^+, proving they are strictly more expressive than Weisfeiler–Lehman colorings in theory and achieving superior performance on graph classification and regression benchmarks, including bioinformatics, social networks, and QM9. They further demonstrate stability across data splits and provide an integration mechanism to combine coloring and topological information, offering a practical pathway to richer graph representations with theoretical guarantees.

Abstract

While message passing graph neural networks result in informative node embeddings, they may suffer from describing the topological properties of graphs. To this end, node filtration has been widely used as an attempt to obtain the topological information of a graph using persistence diagrams. However, these attempts have faced the problem of losing node embedding information, which in turn prevents them from providing a more expressive graph representation. To tackle this issue, we shift our focus to edge filtration and introduce a novel edge filtration-based persistence diagram, named Topological Edge Diagram (TED), which is mathematically proven to preserve node embedding information as well as contain additional topological information. To implement TED, we propose a neural network based algorithm, named Line Graph Vietoris-Rips (LGVR) Persistence Diagram, that extracts edge information by transforming a graph into its line graph. Through LGVR, we propose two model frameworks that can be applied to any message passing GNNs, and prove that they are strictly more powerful than Weisfeiler-Lehman type colorings. Finally we empirically validate superior performance of our models on several graph classification and regression benchmarks.

Figures (9)

• Figure 1: An overview of node filtration and edge filtration of WL coloring. For graphs $G$ and $H$ distinguishable by WL test, the edge filtration-based persistence diagram can also distinguish them while the node filtration-based one cannot.
• Figure 2: The difference in subgraph construction between two types of filtrations: node filtration and edge filtration.
• Figure 3: An example of a pair of non-isomorphic graphs that are not distinguishable by WL test.
• Figure 4: An overall framework of Algorithm \ref{['algo:lgvr_algo']}. Note that the map $t_{\phi}$ transforms a colored graph $(G, \mathcal{C})$ into a colored line graph $(L_{G}, \mathcal{C}^{\phi})$ as described in Construction of the map $t_{\phi}$.
• Figure 5: Two model frameworks: $\mathcal{C}$-${\rm LGVR}$ and $\mathcal{C}$-${\rm LGVR}^{+}$. Note that ENC component refers to the process of extracting node coloring from the message passing graph neural network (or coloring) $\mathcal{C}$ as described in Section \ref{['subsection:model_architectures']} and \ref{['subsection:model_expressivity']}.

Theorems & Definitions (18)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Remark 5
• Definition 6
• Definition 7: Degree Assumption
• Lemma 8
• Theorem 9
• Definition 10