Centered Triangle Closure Regularizer

Integrate the regularizer after the model produces a soft affinity matrix. In a graph transformer, compute attention logits L=QK^T/sqrt(h), symmetrize them with L_sym=(L+L^T)/2, and define P=sigmoid(L_sym+M), where M is the graph mask. In a GNN, use the learned edge scorer as P directly. Compute p_theta as the mean of P_ij over all distinct ordered vertex pairs, then set X_ij=(P_ij-p_theta)/sqrt(p_theta(1-p_theta)+epsilon). For small graphs, compute S_triangle(P)=sum_{i<j<k} X_ij X_ik X_jk exactly, or equivalently use trace(X^3)/6 when X is symmetric with zero diagonal. For larger graphs, sample B distinct vertex triples uniformly and use the unbiased estimate S_hat=(n choose 3)/B times the sum over sampled triples of X_ij X_ik X_jk. Optimize task_loss - lambda*S_hat/(n choose 3) + mu*(p_theta-p_target)^2. Detach neither X nor p_theta so gradients affect affinities, but optionally stop gradients through p_theta for the first warmup epochs. The paper provides the centering and motif statistic; choose lambda, mu, and p_target empirically. First test a two-layer GCN and a small graph transformer on Cora and on synthetic points sampled from a low-dimensional sphere, comparing cross-entropy alone, uncentered triangle regularization, and this centered version. Measure validation accuracy, calibration, and performance after random edge rewiring at fixed density. Success is improved robustness or accuracy at equal parameter count without affinity collapse or density drift.