How Expressive are Knowledge Graph Foundation Models?

Knowledge Graph Foundation Models (KGFMs) are at the frontier for deep learning on knowledge graphs (KGs), as they can generalize to completely novel knowledge graphs with different relational vocabularies. Despite their empirical success, our theoretical understanding of KGFMs remains very limited. In this paper, we conduct a rigorous study of the expressive power of KGFMs. Specifically, we show that the expressive power of KGFMs directly depends on the motifs that are used to learn the relation representations. We then observe that the most typical motifs used in the existing literature are binary, as the representations are learned based on how pairs of relations interact, which limits the model's expressiveness. As part of our study, we design more expressive KGFMs using richer motifs, which necessitate learning relation representations based on, e.g., how triples of relations interact with each other. Finally, we empirically validate our theoretical findings, showing that the use of richer motifs results in better performance on a wide range of datasets drawn from different domains.

Explaining k-Nearest Neighbors: Abductive and Counterfactual Explanations

Despite the wide use of $k$-Nearest Neighbors as classification models, their explainability properties remain poorly understood from a theoretical perspective. While nearest neighbors classifiers offer interpretability from a "data perspective", in which the classification of an input vector $\bar{x}$ is explained by identifying the vectors $\bar{v}_1, \ldots, \bar{v}_k$ in the training set that determine the classification of $\bar{x}$, we argue that such explanations can be impractical in high-dimensional applications, where each vector has hundreds or thousands of features and it is not clear what their relative importance is. Hence, we focus on understanding nearest neighbor classifications through a "feature perspective", in which the goal is to identify how the values of the features in $\bar{x}$ affect its classification. Concretely, we study abductive explanations such as "minimum sufficient reasons", which correspond to sets of features in $\bar{x}$ that are enough to guarantee its classification, and "counterfactual explanations" based on the minimum distance feature changes one would have to perform in $\bar{x}$ to change its classification. We present a detailed landscape of positive and negative complexity results for counterfactual and abductive explanations, distinguishing between discrete and continuous feature spaces, and considering the impact of the choice of distance function involved. Finally, we show that despite some negative complexity results, Integer Quadratic Programming and SAT solving allow for computing explanations in practice.

Link Prediction with Relational Hypergraphs

Link prediction with knowledge graphs has been thoroughly studied in graph machine learning, leading to a rich landscape of graph neural network architectures with successful applications. Nonetheless, it remains challenging to transfer the success of these architectures to link prediction with relational hypergraphs. The presence of relational hyperedges makes link prediction a task between $k$ nodes for varying choices of $k$, which is substantially harder than link prediction with knowledge graphs, where every relation is binary ($k=2$). In this paper, we propose two frameworks for link prediction with relational hypergraphs and conduct a thorough analysis of the expressive power of the resulting model architectures via corresponding relational Weisfeiler-Leman algorithms, and also via some natural logical formalisms. Through extensive empirical analysis, we validate the power of the proposed model architectures on various relational hypergraph benchmarks. The resulting model architectures substantially outperform every baseline for inductive link prediction, and lead to state-of-the-art results for transductive link prediction. Our study therefore unlocks applications of graph neural networks to fully relational structures.

A Theory of Link Prediction via Relational Weisfeiler-Leman

Graph neural networks are prominent models for representation learning over graph-structured data. While the capabilities and limitations of these models are well-understood for simple graphs, our understanding remains highly incomplete in the context of knowledge graphs. The goal of this work is to provide a systematic understanding of the landscape of graph neural networks for knowledge graphs pertaining the prominent task of link prediction. Our analysis entails a unifying perspective on seemingly unrelated models, and unlocks a series of other models. The expressive power of various models is characterized via a corresponding relational Weisfeiler-Leman algorithm with different initialization regimes. This analysis is extended to provide a precise logical characterization of the class of functions captured by a class of graph neural networks. Our theoretical findings explain the benefits of some widely employed practical design choices, which are validated empirically.

Weisfeiler and Leman Go Relational

Knowledge graphs, modeling multi-relational data, improve numerous applications such as question answering or graph logical reasoning. Many graph neural networks for such data emerged recently, often outperforming shallow architectures. However, the design of such multi-relational graph neural networks is ad-hoc, driven mainly by intuition and empirical insights. Up to now, their expressivity, their relation to each other, and their (practical) learning performance is poorly understood. Here, we initiate the study of deriving a more principled understanding of multi-relational graph neural networks. Namely, we investigate the limitations in the expressive power of the well-known Relational GCN and Compositional GCN architectures and shed some light on their practical learning performance. By aligning both architectures with a suitable version of the Weisfeiler-Leman test, we establish under which conditions both models have the same expressive power in distinguishing non-isomorphic (multi-relational) graphs or vertices with different structural roles. Further, by leveraging recent progress in designing expressive graph neural networks, we introduce the $k$-RN architecture that provably overcomes the expressiveness limitations of the above two architectures. Empirically, we confirm our theoretical findings in a vertex classification setting over small and large multi-relational graphs.