Sam Adam-Day (he/him)

Graph neural networks (GNNs) are the predominant architectures for a variety of learning tasks on graphs. We present a new angle on the expressive power of GNNs by studying how the predictions of a GNN probabilistic classifier evolve as we apply it on larger graphs drawn from some random graph model. We show that the output converges to a constant function, which upper-bounds what these classifiers can express uniformly. This convergence phenomenon applies to a very wide class of GNNs, including state of the art models, with aggregates including mean and the attention-based mechanism of graph transformers. Our results apply to a broad class of random graph models, including the (sparse) Erdős-Rényi model and the stochastic block model. We empirically validate these findings, observing that the convergence phenomenon already manifests itself on graphs of relatively modest size.

Graph neural networks (GNNs) are de facto standard deep learning architectures for machine learning on graphs. This has led to a large body of work analyzing the capabilities and limitations of these models, particularly pertaining to their representation and extrapolation capacity. We offer a novel theoretical perspective on the representation and extrapolation capacity of GNNs, by answering the question: how do GNNs behave as the number of graph nodes become very large? Under mild assumptions, we show that when we draw graphs of increasing size from the Erdős-Rényi model, the probability that such graphs are mapped to a particular output by a class of GNN classifiers tends to either zero or to one. This class includes the popular graph convolutional network architecture. The result establishes ’zero-one laws’ for these GNNs, and analogously to other convergence laws, entails theoretical limitations on their capacity. We empirically verify our results, observing that the theoretical asymptotic limits are evident already on relatively small graphs.

We investigate a recently-devised polyhedral semantics for intermediate logics, in which formulas are interpreted in n-dimensional polyhedra. An intermediate logic is polyhedrally complete if it is complete with respect to some class of polyhedra. The first main result of this paper is a necessary and sufficient condition for the polyhedral-completeness of a logic. This condition, which we call the Nerve Criterion, is expressed in terms of Alexandrov’s notion of the nerve of a poset. It affords a purely combinatorial characterisation of polyhedrally-complete logics. Using the Nerve Criterion we show, easily, that there are continuum many intermediate logics that are not polyhedrally-complete but which have the finite model property. We also provide, at considerable combinatorial labour, a countably infinite class of logics axiomatised by the Jankov-Fine formulas of ’starlike trees’ all of which are polyhedrally-complete. The polyhedral completeness theorem for these ’starlike logics’ is the second main result of this paper.