WL Tests Are Far from All We Need: Revisiting WL-Test Hardness and GNN Expressive Power from a Distributed Computation Perspective

TL;DR

This work questions the WL-based lens on GNN expressivity by adopting a distributed-computation perspective. It delivers near-tight bounds on the depth $d$, width $w$, and precision $p$ required to simulate a single WL iteration under deterministic, Las Vegas, and bounded-error randomized regimes, demonstrating that WL iterations are not always primitive for constant-depth GNNs. The authors introduce a generalized Resource-Limited CONGEST (RL-CONGEST) framework with an explicit preprocessing phase to standardize expressivity analyses and quantify how preprocessing choices, including virtual nodes and edges, influence WL behavior. They further show that certain preprocessing strategies can inject implicit shortcuts, while other configurations can meaningfully alter the depth-width-precision requirements, offering nuanced guidance for studying GNN expressivity beyond WL equivalence and for designing GNN architectures under resource constraints.

Abstract

The expressive power of graph neural networks (GNNs) is often studied through their relationship to the Weisfeiler-Lehman (WL) tests. Despite its influence, this perspective leaves two gaps: (i) it is unclear whether WL tests are sufficiently primitive for understanding GNN expressivity, and (ii) WL-induced equivalence does not align well with characterizing the function classes that GNNs can approximate or compute. We attempt to address both gaps. First, we strengthen hardness results for the vanilla WL test, showing that in many settings it is not primitive enough to be implemented by constant-depth GNNs. Second, we propose an alternative framework for studying GNN expressivity based on an extended CONGEST model with an explicit preprocessing phase. Within this framework, we identify implicit shortcuts introduced in prior analyses and establish further results for WL tests in settings where graphs are augmented with virtual nodes and virtual edges.

Figures (5)

• Figure 1: The computation of a GNN model in our expressive-power analysis framework based on RL-CONGEST. Left: Given an attributed graph $G$ with input features $\boldsymbol{X}$, preprocessing constructs a new graph $G'$ with features $\boldsymbol{X}'$. Middle: In each round, every node (1) receives messages from its neighbors, (2) performs local computation within the complexity class $\textsf{C}$, and (3) sends messages to its neighbors. Right: Postprocessing produces the final output for the downstream task.
• Figure 2: The constructed basic graph $G_{(n, m)}$. Nodes are colored according to $\boldsymbol{x}$.
• Figure 3: The constructed basic graph $G_{n}$.
• Figure 4: The constructed basic graph $G_{n}$.
• Figure 5: A path graph and a cycle graph that differ only by the edge $(0, n - 1)$, affecting biconnectivity.

Theorems & Definitions (43)

• Definition 1: CONGEST Model
• Definition 2: Weisfeiler-Lehman Relation
• Theorem 1
• Theorem 2
• Theorem 3: cf. Section 3 of aamand2022exponentially
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Definition 3: Expressivity Analysis Framework Based on RL-CONGEST