Rank-Two Motif Spectral Architecture Library

Implement a motif mixer as a sparse graph layer over a set of directed edge channels. Choose a small candidate set, for example figure-eight (a,b) in {(1,1),(1,2),(2,2)}, theta (a,b,c) in {(1,1,1),(1,1,2),(1,2,2)}, and dumbbell (a,b,c) in {(1,1,1),(1,2,1),(2,2,1)}. For each candidate, instantiate its directed non-backtracking transition matrix B_m and a separate linear map W_m. Compute alpha_m offline by bisection over x in [1+1e-6, 10] using the corresponding displayed equation; increase the upper bound until the polynomial changes sign. Normalize each branch as u_m = B_m z W_m / alpha_m, then combine u = sum_m softmax(beta)_m u_m, where beta are learned architecture logits. Pseudocode: for m in motifs: alpha[m]=root_equation(m); z_m=NBProp(B_m,z); z_m=z_m/alpha[m]; u += softmax(beta)[m]*Linear_m(z_m); h=Update(h,Aggregate(u)). The paper supplies exact alpha values for the ideal motif; on arbitrary input graphs, estimate the actual rho(B_m) by 15-step power iteration and optionally use alpha_eff=(1-lambda)*alpha_m+lambda*rho_hat. Measure computed math versus empirical behavior separately: root solving gives the designed entropy, while validation loss and feature norms test whether that entropy is useful. First cheap experiment: train 2-4 layer models on synthetic stochastic block graphs and molecular TU datasets, comparing a standard GIN/GCN with a parameter-matched motif mixer. Report accuracy versus depth, activation amplification, and sparse FLOPs. A useful signal is that the learned beta selects different motifs by graph family and achieves equal accuracy with fewer layers or fewer edges processed.