Exponentially Improving the Complexity of Simulating the Weisfeiler-Lehman Test with Graph Neural Networks

Anders Aamand, Justin Y Chen, Piotr Indyk, Shyam Narayanan, Ronitt Rubinfeld, Nicholas Schiefer, Sandeep Silwal, Tal Wagner

Keywords: graph neural network, Weisfeiler-Lehman, graph isomorphism, expressivity

TL;DR: We give an *efficient* construction of GNNs simulating the Weisfeiler-Lehman test for graph isomorphism, which has been recently shown to capture their expressiveness. Our GNN size is exponentially better than previous ones..

Abstract:

Recent work shows that the expressive power of Graph Neural Networks (GNNs) in distinguishing non-isomorphic graphs is exactly the same as that of the Weisfeiler-Lehman (WL) graph test. In particular, they show that the WL test can be simulated by GNNs. However, those simulations involve neural networks for the “combine” function of size polynomial or even exponential in the number of graph nodes $n$, as well as feature vectors of length linear in $n$.

We present an improved simulation of the WL test on GNNs with {\em exponentially} lower complexity. In particular, the neural network implementing the combine function in each node has only $\mathrm{polylog}(n)$ parameters, and the feature vectors exchanged by the nodes of GNN consists of only $O(\log n)$ bits. We also give logarithmic lower bounds for the feature vector length and the size of the neural networks, showing the (near)-optimality of our construction.