We study how information propagates in decoder-only Transformers, which are the architectural foundation of most existing frontier large language models (LLMs). We rely on a theoretical signal propagation analysis---specifically, we analyse the representations of the last token in the final layer of the Transformer, as this is the representation used for next-token prediction. Our analysis reveals a representational collapse phenomenon: we prove that certain distinct pairs of inputs to the Transformer can yield arbitrarily close representations in the final token. This effect is exacerbated by the low-precision floating-point formats frequently used in modern LLMs. As a result, the model is provably unable to respond to these sequences in different ways---leading to errors in, e.g., tasks involving counting or copying.

Information over-squashing is a phenomenon of inefficient information propagation between distant nodes on networks. It is an important problem that is known to significantly impact the training of graph neural networks (GNNs), as the receptive field of a node grows exponentially. To mitigate this problem, a preprocessing procedure known as rewiring is often applied to the input network. In this paper, we investigate the use of discrete analogues of classical geometric notions of curvature to model information flow on networks and rewire them. We show that these classical notions achieve state-of-the-art performance in GNN training accuracy on a variety of real-world network datasets. Moreover, compared to the current state-of-the-art, these classical notions exhibit a clear advantage in computational runtime by several orders of magnitude.