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The expressivity of Graph Neural Networks (GNNs) has been studied broadly inrecent years to reveal the design principles for more powerful GNNs. Graphcanonization is known as a typical approach to distinguish non-isomorphicgraphs, yet rarely adopted when developing expressive GNNs. This paper proposesto maximize the expressivity of GNNs by graph canonization, then the power ofsuch GNNs is studies from the perspective of model stability. A stable GNN willmap similar graphs to close graph representations in the vectorial space, andthe stability of GNNs is critical to generalize their performance to unseengraphs. We theoretically reveal the trade-off of expressivity and stability ingraph-canonization-enhanced GNNs. Then we introduce a notion of universal graphcanonization as the general solution to address the trade-off and characterizea widely applicable sufficient condition to solve the universal graphcanonization. A comprehensive set of experiments demonstrates the effectivenessof the proposed method. In many popular graph benchmark datasets, graphcanonization successfully enhances GNNs and provides highly competitiveperformance, indicating the capability and great potential of proposed methodin general graph representation learning. In graph datasets where thesufficient condition holds, GNNs enhanced by universal graph canonizationconsistently outperform GNN baselines and successfully improve the SOTAperformance up to $31\%$, providing the optimal solution to numerouschallenging real-world graph analytical tasks like gene network representationlearning in bioinformatics.

Rethinking the Power of Graph Canonization in Graph Representation Learning with Stability