Juan L. Reutter

Floris Geerts, Juan L. Reutter

Characterizing the separation power of graph neural networks (GNNs) provides an understanding of their limitations for graph learning tasks. Results regarding separation power are, however, usually geared at specific GNN architectures, and tools for understanding arbitrary GNN architectures are generally lacking. We provide an elegant way to easily obtain bounds on the separation power of GNNs in terms of the Weisfeiler-Leman (WL) tests, which have become the yardstick to measure the separation power of GNNs. The crux is to view GNNs as expressions in a procedural tensor language describing the computations in the layers of the GNNs. Then, by a simple analysis of the obtained expressions, in terms of the number of indexes and the nesting depth of summations, bounds on the separation power in terms of the WL-tests readily follow. We use tensor language to define Higher-Order Message-Passing Neural Networks (or k-MPNNs), a natural extension of MPNNs. Furthermore, the tensor language point of view allows for the derivation of universality results for classes of GNNs in a natural way. Our approach provides a toolbox with which GNN architecture designers can analyze the separation power of their GNNs, without needing to know the intricacies of the WL-tests. We also provide insights in what is needed to boost the separation power of GNNs.

Tamara Cucumides, Daniel Daza, Pablo Barceló et al.

The challenge of answering graph queries over incomplete knowledge graphs is gaining significant attention in the machine learning community. Neuro-symbolic models have emerged as a promising approach, combining good performance with high interpretability. These models utilize trained architectures to execute atomic queries and integrate modules that mimic symbolic query operators. However, most neuro-symbolic query processors are constrained to tree-like graph pattern queries. These queries admit a bottom-up execution with constant values or anchors at the leaves and the target variable at the root. While expressive, tree-like queries fail to capture critical properties in knowledge graphs, such as the existence of multiple edges between entities or the presence of triangles. We introduce a framework for answering arbitrary graph pattern queries over incomplete knowledge graphs, encompassing both cyclic queries and tree-like queries with existentially quantified leaves. These classes of queries are vital for practical applications but are beyond the scope of most current neuro-symbolic models. Our approach employs an approximation scheme that facilitates acyclic traversals for cyclic patterns, thereby embedding additional symbolic bias into the query execution process. Our experimental evaluation demonstrates that our framework performs competitively on three datasets, effectively handling cyclic queries through our approximation strategy. Additionally, it maintains the performance of existing neuro-symbolic models on anchored tree-like queries and extends their capabilities to queries with existentially quantified variables.

Xingyue Huang, Pablo Barceló, Michael M. Bronstein et al. · deepmind

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.

Pablo Barceló, Floris Geerts, Juan Reutter et al.

Various recent proposals increase the distinguishing power of Graph Neural Networks GNNs by propagating features between $k$-tuples of vertices. The distinguishing power of these "higher-order'' GNNs is known to be bounded by the $k$-dimensional Weisfeiler-Leman (WL) test, yet their $\mathcal O(n^k)$ memory requirements limit their applicability. Other proposals infuse GNNs with local higher-order graph structural information from the start, hereby inheriting the desirable $\mathcal O(n)$ memory requirement from GNNs at the cost of a one-time, possibly non-linear, preprocessing step. We propose local graph parameter enabled GNNs as a framework for studying the latter kind of approaches and precisely characterize their distinguishing power, in terms of a variant of the WL test, and in terms of the graph structural properties that they can take into account. Local graph parameters can be added to any GNN architecture, and are cheap to compute. In terms of expressive power, our proposal lies in the middle of GNNs and their higher-order counterparts. Further, we propose several techniques to aide in choosing the right local graph parameters. Our results connect GNNs with deep results in finite model theory and finite variable logics. Our experimental evaluation shows that adding local graph parameters often has a positive effect for a variety of GNNs, datasets and graph learning tasks.