As a powerful framework for graph representation learning, Graph Neural Networks (GNNs) have garnered significant attention in recent years. However, to the best of our knowledge, there has been no formal analysis of the logical expressiveness of GNNs as Boolean node classifiers over multi-relational graphs, where each edge carries a specific relation type. In this paper, we investigate $\mathcal{FOC}_2$, a fragment of first-order logic with two variables and counting quantifiers. On the negative side, we demonstrate that the R$^2$-GNN architecture, which extends the local message passing GNN by incorporating global readout, fails to capture $\mathcal{FOC}_2$ classifiers in the general case. Nevertheless, on the positive side, we establish that R$^2$-GNNs models are equivalent to $\mathcal{FOC}_2$ classifiers under certain restricted yet reasonable scenarios. To address the limitations of R$^2$-GNNs regarding expressiveness, we propose a simple graph transformation technique, akin to a preprocessing step, which can be executed in linear time. This transformation enables R$^2$-GNNs to effectively capture any $\mathcal{FOC}_2$ classifiers when applied to the transformed input graph. Moreover, we extend our analysis of expressiveness and graph transformation to temporal graphs, exploring several temporal GNN architectures and providing an expressiveness hierarchy for them. To validate our findings, we implement R$^2$-GNNs and the graph transformation technique and conduct empirical tests in node classification tasks against various well-known GNN architectures that support multi-relational or temporal graphs.

As a powerful framework for graph representation learning, Graph Neural Networks (GNNs) have garnered significant attention in recent years. However, to the best of our knowledge, there has been no formal analysis of the logical expressiveness of GNNs as Boolean node classifiers over multi-relational graphs, where each edge carries a specific relation type. In this paper, we investigate \mathcal{FOC}_2, a fragment of first-order logic with two variables and counting quantifiers. On the negative side, we demonstrate that the R 2 -GNN architecture, which extends the local message passing GNN by incorporating global readout, fails to capture \mathcal{FOC}_2 classifiers in the general case. Nevertheless, on the positive side, we establish that R 2 -GNNs models are equivalent to \mathcal{FOC}_2 classifiers under certain restricted yet reasonable scenarios. To address the limitations of R 2 -GNNs regarding expressiveness, we propose a simple graph transformation technique, akin to a preprocessing step, which can be executed in linear time.

Graph Neural Networks (GNNs) have been recently introduced to learn from knowledge graph (KG) and achieved state-of-the-art performance in KG reasoning. However, a theoretical certification for their good empirical performance is still absent. Besides, while logic in KG is important for inductive and interpretable inference, existing GNN-based methods are just designed to fit data distributions with limited knowledge of their logical expressiveness. We propose to fill the above gap in this paper. Specifically, we theoretically analyze GNN from logical expressiveness and find out what kind of logical rules can be captured from KG. Our results first show that GNN can capture logical rules from graded modal logic, providing a new theoretical tool for analyzing the expressiveness of GNN for KG reasoning; and a query labeling trick makes it easier for GNN to capture logical rules, explaining why SOTA methods are mainly based on labeling trick. Finally, insights from our theory motivate the development of an entity labeling method for capturing difficult logical rules. Experimental results are consistent with our theoretical results and verify the effectiveness of our proposed method.