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

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.