Numerical and effective rank of node representations collapse toward 1 with depth in real trained GCN/GAT networks, and correlate with test accuracy far better than Dirichlet energy

Numerical Rank and Effective Rank of the last-hidden-layer node-representation matrix decline sharply as depth increases from 2 to 24 layers in real trained GCN and GAT networks, e.g. dropping from an effective rank of about 1084 to 13.6 on Cora as depth grows [zhang-etal-2025-are-we-measuring-oversmoothing-in-gnns-correctly] Across homophilic (Cora, Citeseer, Pubmed), heterophilic (Squirrel, Chameleon, Amazon Ratings), and large-scale (OGB-Arxiv) real benchmark graphs, rank-based metrics correlate with test accuracy far better than Dirichlet energy (e.g. Cora/GCN: NumRank correlation 0.59, Erank 0.97, vs. Dirichlet energy -0.79) [zhang-etal-2025-are-we-measuring-oversmoothing-in-gnns-correctly] Theorem-level result proves numerical rank converges to 1 for a broad family of GNN architectures as depth grows, paired with the empirical measurement (10 independently trained networks per depth/dataset configuration) rather than left purely theoretical [zhang-etal-2025-are-we-measuring-oversmoothing-in-gnns-correctly]