Linear GCNs Need Better Bias, Not More Expressive Power

Do modern GNNs succeed because of architectural expressive power or because of implicit bias? We show that simple linear graph convolutions with sum aggregation and MLP classifiers can match the generalization performance of state-of-the-art graph learning methods. Theoretically, sum aggregation after a linear feature transformation can preserve neighborhood multiset information, yielding Weisfeiler–Lehman-level expressive power. Empirically, however, this potential is realized only through distillation from more complex GNN teachers. When trained from scratch, linear GCNs often fail to generalize. This reveals a gap between what simple models can express and what standard training selects. Our findings suggest that progress in graph learning may depend less on increasing expressiveness and more on controlling implicit bias, trainability, and model selection. Our experimental design provides a framework to distinguish these concepts.