Volume-Mass Diffusion GNN

Integrate the mechanism into the propagation block of a GCN or graph diffusion network. Given sparse adjacency \(\mathbf A\), calculate \(k_i\), form \(\mathbf L=\mathbf D-\mathbf A\), and choose positive volumes using fixed metadata, \(v_i\propto k_i^\alpha\), or learned values \(v_i=\operatorname{softplus}(q_i)+\epsilon\). Apply one or more diffusion substeps before the learned feature transform. Pseudocode is: construct sparse \(L=D-A\); define the operator \(z\mapsto V^{-1/2}L V^{-1/2}z\); estimate \(\hat\lambda_{\max}\) with 5--10 power iterations; set \(\eta=\gamma 2/(\hat\lambda_{\max}+\epsilon)\), where \(0<\gamma<1\); update \(H\leftarrow H-\eta V^{-1}LH\); then compute \(H\leftarrow\sigma(HW)\). Add a residual connection if desired. The paper supplies the generalized diffusion operator, similarity transform, and stability ceiling; the implementation estimates the largest eigenvalue, learned volumes, and localization statistics. First test on a small stochastic-block-model graph and Cora against a standard GCN and normalized GCN with equal parameter counts. Sweep \(v_i=k_i^\alpha\) and lognormal volume noise with standard deviation \(s\). The predicted signature is an explosive feature-norm boundary at \(\eta_c\approx2/\lambda_{\max}(S)\), with measured threshold within 20 percent of this estimate. Also measure the top-eigenvector IPR: it should increase with \(\operatorname{std}(\log r_i)\), and its localization center should shift away from the highest-degree node when degree-volume correlations are disrupted.