The logic of rational graph neural networks
Keywords: Graph Neural Networks, Rational activations, Expressivity, Logic
TL;DR: This article article investigates the expressive power of Graph Neural Networks with rational activations.
Abstract: The expressivity of Graph Neural Networks (GNNs) can be described via appropriate fragments of the first-order logic. In this context, uniform expressivity guarantees that a GNN can express a logical query without the parameters depending on the size of the input graphs. It has been established that the two-variable guarded fragment with counting (GC2) can be expressed uniformly via Rectified Linear Unit (ReLU) GNNs [Barcelo &. Al., 2020]. Moreover, GC2 is the fragment that can be expressed at most by a GNN with any activation function. In this article, we prove that, on the contrary of ReLU GNNs, there are GC2 queries that cannot be uniformly expressed via any GNN with rational activations. As a consequence, non-polynomial activation functions do not grant GNNs GC2 uniform expressivity in general, answering an open question formulated by [Grohe, 2021]. We then present a strict subfragment of GC2 (RGC2), and prove that rational GNNs can express RGC2 queries uniformly over all graphs. Our numerical experiments illustrates that despite this theoretical disadvantage, rational GNNs are still able to learn some GC2 queries if some level of error is allowed.
Primary Area: learning on graphs and other geometries & topologies
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Submission Number: 7943
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