Halting Recurrent GNNs and the Graded $μ$-Calculus

Graph Neural Networks (GNNs) are a class of machine-learning models that operate on graph-structured data. Their expressive power is intimately related to logics that are invariant under graded bisimilarity. Current proposals for recurrent GNNs either assume that the graph size is given to the model, or suffer from a lack of termination guarantees. In this paper, we propose a halting mechanism for recurrent GNNs. We prove that our halting model can express all node classifiers definable in graded modal mu-calculus, even for the standard GNN variant that is oblivious to the graph size. A recent breakthrough in the study of the expressivity of graded modal mu-calculus in the finite suggests that conversely, restricted to node classifiers definable in monadic second-order logic, recurrent GNNs can express only node classifiers definable in graded modal mu-calculus. To prove our main result, we develop a new approximate semantics for graded mu-calculus, which we believe to be of independent interest. We leverage this new semantics into a new model-checking algorithm, called the counting algorithm, which is oblivious to the graph size. In a final step we show that the counting algorithm can be implemented on a halting recurrent GNN.

Learning Graph Neural Networks using Exact Compression

Graph Neural Networks (GNNs) are a form of deep learning that enable a wide range of machine learning applications on graph-structured data. The learning of GNNs, however, is known to pose challenges for memory-constrained devices such as GPUs. In this paper, we study exact compression as a way to reduce the memory requirements of learning GNNs on large graphs. In particular, we adopt a formal approach to compression and propose a methodology that transforms GNN learning problems into provably equivalent compressed GNN learning problems. In a preliminary experimental evaluation, we give insights into the compression ratios that can be obtained on real-world graphs and apply our methodology to an existing GNN benchmark.