Depth, Normalization, and Graph Structure in Vanilla GCNs: A Controlled Empirical Study

Abstract Prior work offers contradictory guidance on whether normalization mitigates oversmoothing in deep Graph Convolutional Networks. We revisit this question under strict experimental control—vanilla GCN, fully fixed hyperparameters, no architectural augmentations—across six normalization schemes, six node-classification benchmarks, and depths 1–20 (3600 runs total). Three empirical observations emerge. (O1) Normalization effects concentrate at the $${{\varvec{L}}}~{{\textbf {\!:\!}}}~{\mathbf {1\!\rightarrow \!2}}$$ L : 1 → 2 transition and show diminishing returns beyond depth 3; at $${{\varvec{L}}} {\mathbf { = 1}}$$ L = 1 normalization is structurally inoperative and all schemes are identical. (O2) At depth $$\ge $$ ≥ 6, graph topology governs the accuracy trajectory: homophily is the only statistically significant predictor of depth-sensitivity ( $${{\varvec{r}}} = {\mathbf {-0.87}}$$ r = - 0.87 , $${{\varvec{p}}}~{\mathbf {=~0.02}}$$ p = 0.02 ), while spectral gap, degree, and clustering are non-significant in our dataset family. (O3) Normalization reduces cross-seed training variance by up to 90% at $${{\varvec{L}}} {\mathbf { = 2}}$$ L = 2 on dense graphs but does not prevent eventual collapse. Depth-20 results confirm continued degradation on all datasets (Cora: $${\mathbf {0.75}}{\mathbf { \rightarrow }}{\mathbf {0.57}}{\mathbf { \rightarrow }}{\mathbf {0.31}}$$ 0.75 → 0.57 → 0.31 ). Wilcoxon signed-rank tests at depth 10 show BatchNorm and GraphNorm significantly outperform NoNorm ( $${{\varvec{p}}}~{\mathbf {<~0.001}}$$ p < 0.001 ), while NodeNorm significantly underperforms ( $${{\varvec{p}}}~{\mathbf {<~0.001}}$$ p < 0.001 ). These findings reframe normalization as an optimization stabilizer at shallow-to-moderate depth, not a remedy for propagation-driven oversmoothing, and provide a reproducible stress-test protocol for evaluating depth-robust GNN mechanisms.