Exponentially Improving the Complexity of Simulating the WeisfeilerLehman Test with Graph Neural Networks
Abstract
Recent work shows that the expressive power of Graph Neural Networks (GNNs) in distinguishing nonisomorphic graphs is exactly the same as that of the WeisfeilerLehman (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 \emph{exponentially} lower complexity. In particular, the neural network implementing the combine function in each node has only a polylogarithmic number of parameters in $n$, 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2022
 DOI:
 10.48550/arXiv.2211.03232
 arXiv:
 arXiv:2211.03232
 Bibcode:
 2022arXiv221103232A
 Keywords:

 Computer Science  Machine Learning;
 Statistics  Machine Learning
 EPrint:
 22 pages,5 figures, published at NeurIPS 2022. Updated funding statements