ADAPTIVEMIXGNN: Local Adaptive Inductive Bias for Heterophilic Node Classification

Most GNNs apply a uniform global filter to every node, implicitly assuming one dominant structural regime. Real graphs violate this assumption: homophilic and heterophilic patterns coexist and vary locally. We introduce AdaptiveMixGNN, a first-order spectral GNN that preserves scale and simplicity by learning a per-node mixing between low-pass and high-pass shifts: $$ S_{\alpha} = \mathrm{diag}(\alpha)\, S_{\mathrm{LP}} + (I - \mathrm{diag}(\alpha))\, S_{\mathrm{HP}}, $$ with $$ \alpha_i = \sigma(h_i^{\top}\theta + b). $$ This adds only d+1 parameters per layer and keeps O(L · |E|) complexity, comparable to GCNs. On heterophilic benchmarks, AdaptiveMixGNN reaches 79.46% accuracy on Texas and 79.61% on Wisconsin, outperforming the polynomial filters we evaluated (K ≥ 10) while avoiding their overfitting pathologies on small graphs. Ablation studies show that node-wise adaptivity acts as an insurance policy against catastrophic failures of fixed filters, with gains of up to +10.59% over the best static baseline. Finally, a per-node homophily analysis links the learned α values to local label structure (Texas: mean homophily 0.033 versus 0.247 for correct versus incorrect nodes), suggesting that the model discovers a meaningful local frequency response.