Incidence-Mixed Simplicial Diffusion

Integrate the mechanism as an edge-level message-passing block before pooling edge features to nodes or to a graph representation. For every input graph, construct sparse B1 and B2. Compute the vertex channel by multiplying B1 transpose, the inverse diagonal vertex-degree matrix, and B1, then row-normalize the resulting sparse matrix. Compute the triangle channel from B2 times B2 transpose and row-normalize it. Add self-loops before normalization so edges with no retained triangles remain numerically valid. Set P = q PV + (1 minus q) PT and return H next = (1 minus alpha) H + alpha P H Theta. Begin with q = 0.5, alpha = 0.5, and a shared Theta; later test a learned q using q = sigmoid(a), or a graph-dependent q predicted from pooled features. The incidence matrices, degrees, and sparse propagation are computed exactly. Estimate mu2 with 10 to 20 Lanczos iterations and estimate the stationary distribution pi by repeatedly applying P to a probability vector. The first cheap experiment should generate 100-vertex complexes with candidate triangles independently retained at densities p from 0.05 to 0.95. Assign edge labels using a latent triangle-dependent rule and compare an ordinary edge GCN with this layer at matched parameter count. Measure validation accuracy, mu2, tau, IPR, and feature autocorrelation decay. The predicted signature is an interior maximum of mu2, tau, and IPR at intermediate p, while empirical diffusion decay should match absolute value of mu2 to the power t within approximately 20 percent.