How hard is graph isomorphism for graph neural networks?
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A hallmark of graph neural networks is their ability to distinguish the isomorphism class of their inputs. This study derives the first hardness results for graph isomorphism in the message-passing model (MPNN). MPNN encompasses the majority of graph neural networks used today and is universal in the limit when nodes are given unique features. The analysis relies on the introduced measure of communication capacity. Capacity measures how much information the nodes of a network can exchange during the forward pass and depends on the depth, message-size, global state, and width of the architecture. It is shown that the capacity of MPNN needs to grow linearly with the number of nodes so that a network can distinguish trees and quadratically for general connected graphs. Crucially, the derived bounds are applicable not only to worst-case instances but over a portion of all inputs. An empirical study involving 12 tasks of varying difficulty and 420 networks reveals strong alignment between actual performance and theoretical predictions.
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