Rethinking the Expressive Power of GNNs via Graph Biconnectivity

TL;DR

This work introduces graph biconnectivity as a principled lens for understanding GNN expressiveness beyond the WL hierarchy. It shows that most popular GNNs lack the ability to capture biconnectivity metrics, while ESAN/DSS-WL can provably identify certain biconnectivity properties. To bridge practicality and expressiveness, the authors propose Generalized Distance Weisfeiler-Lehman (GD-WL), which encodes distance information into WL refinements, enabling provable expressiveness for all biconnectivity metrics when using SPD-WL (edge-biconnectivity) and RD-WL (vertex-biconnectivity) in tandem. They implement a Transformer-based Graphormer-GD that matches GD-WL in power and demonstrates strong empirical performance on synthetic and real datasets, with favorable parallelizability. The work provides a solid theoretical foundation and a scalable architecture for learning graph properties tied to connectivity, with broad implications for tasks where graph structure is crucial.

Abstract

Designing expressive Graph Neural Networks (GNNs) is a central topic in learning graph-structured data. While numerous approaches have been proposed to improve GNNs in terms of the Weisfeiler-Lehman (WL) test, generally there is still a lack of deep understanding of what additional power they can systematically and provably gain. In this paper, we take a fundamentally different perspective to study the expressive power of GNNs beyond the WL test. Specifically, we introduce a novel class of expressivity metrics via graph biconnectivity and highlight their importance in both theory and practice. As biconnectivity can be easily calculated using simple algorithms that have linear computational costs, it is natural to expect that popular GNNs can learn it easily as well. However, after a thorough review of prior GNN architectures, we surprisingly find that most of them are not expressive for any of these metrics. The only exception is the ESAN framework, for which we give a theoretical justification of its power. We proceed to introduce a principled and more efficient approach, called the Generalized Distance Weisfeiler-Lehman (GD-WL), which is provably expressive for all biconnectivity metrics. Practically, we show GD-WL can be implemented by a Transformer-like architecture that preserves expressiveness and enjoys full parallelizability. A set of experiments on both synthetic and real datasets demonstrates that our approach can consistently outperform prior GNN architectures.

Figures (12)

• Figure 1: An illustration of edge-biconnectivity and vertex-biconnectivity. Cut vertices/edges are outlined in bold red. Gray nodes in (b)/(c) are edge/vertex-biconnected components, respectively.
• Figure 2: Illustration of four representative counterexamples (see \ref{['example:1', 'example:2']} for general definitions). Graphs in the first row have cut vertices (outlined in bold red) and some also have cut edges (denoted as red lines), while graphs in the second row do not have any cut vertex or cut edge.
• Figure 3: Illustration of non-isomorphic distance-regular graphs.
• Figure 4: Illustration of the proof of \ref{['thm:scwl']}. The trees $G_1[{\mathcal{S}}], G_2[g({\mathcal{S}})]$ are outlined by orange.
• Figure 5: Several illustrations to help understand the lemmas.

Theorems & Definitions (128)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5
• Theorem 3.1
• Theorem 3.2
• Theorem 4.1
• Theorem 4.2
• Corollary 4.3