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Message Passing Neural Networks (MPNNs) are instances of Graph NeuralNetworks that leverage the graph to send messages over the edges. Thisinductive bias leads to a phenomenon known as over-squashing, where a nodefeature is insensitive to information contained at distant nodes. Despiterecent methods introduced to mitigate this issue, an understanding of thecauses for over-squashing and of possible solutions are lacking. In thistheoretical work, we prove that: (i) Neural network width can mitigateover-squashing, but at the cost of making the whole network more sensitive;(ii) Conversely, depth cannot help mitigate over-squashing: increasing thenumber of layers leads to over-squashing being dominated by vanishinggradients; (iii) The graph topology plays the greatest role, sinceover-squashing occurs between nodes at high commute (access) time. Our analysisprovides a unified framework to study different recent methods introduced tocope with over-squashing and serves as a justification for a class of methodsthat fall under graph rewiring.

On Over-Squashing in Message Passing Neural Networks: The Impact of Width, Depth, and Topology