The Expressive Power of Random-Feature MPNNs Across Graph Sizes

Key points

• Continuous random node features increase the expressivity of message-passing neural networks (MPNNs), but existing theoretical analyses are not fine-grained enough to capture their limitations, particularly in approximating functions across graph sizes or under a fixed number of random bits.1
• We develop a population-level framework for size-uniform approximation.1
• We use this framework to prove that these MPNNs are not size-uniformly universal, analyze the expressivity gains from colors and independent trials, and establish incomparability between random-color MPNNs and higher-order Weisfeiler-Leman methods.1
• We also give finite-color MPNN constructions for tree subgraph isomorphism via color coding and a local MST task.1

Sources (1)

• [1]The Expressive Power of Random-Feature MPNNs Across Graph Sizes
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 8, 11:38 PM

  For MPNNs with a graph-size-independent Lipschitz bound in their initial features, including standard mean, max, and order-normalized sum architectures, we show that continuous random features can be quantized to finitely many colors with arbitrarily small uniform error in expected output.

  Continuous random node features increase the expressivity of message-passing neural networks (MPNNs), but existing theoretical analyses are not fine-grained enough to capture their limitations, particularly in approximating functions across graph sizes or under a fixed number of random bits.

Extractive summary: sentences quoted from the sources.