Fundamental Limits of Local Graph Neural Networks on High-Girth Graphs

Kalyan Cherukuri

Keywords: Graph neural network, high girth graphs, fundamental limits

Abstract: We determine exact limits of local, message-passing GNNs (Graph neural networks) on graphs of high girth. We prove upper bounds for any GNN with a receptive radius of $L$ on graphs with a maximum degree of $d$ and girth exceeding $2L+1$. The central contribution is a finite-dimensional linear program that characterizes the optimal performance of any $L$-local randomized algorithm on the infinite $d$-regular tree. We demonstrate that this value serves as a hard asymptotic ceiling for local GNNs on large, high-girth graphs and prove the tightness with a GNN construction that achieves this bound.

Submission Number: 31