Gaussian Synergy-Preserving Node Aggregation for Restricted Graph

Undirected weighted graphs model complex systems through pairwise associations, but conventional edge aggregation may fail to preserve the collective dependence between a group of nodes and their shared neighbors. We address this problem through a property-specific graph summarization framework based on a Laplacian-induced Gaussian model. The framework identifies restricted independent-neighbor motifs with a positive mutual-information synergy gap and derives superedge weights that preserve the corresponding dependence during node aggregation. For motifs with one shared neighbor, the solution uniquely preserves the joint dependence. For motifs with two shared neighbors, preservation is guaranteed only under an explicit solvability condition and applies to the two marginal dependences separately. Experiments on synthetic motifs, weighted real networks, and multiple random graph families verify numerical preservation, characterize the applicability of the solvability condition, and evaluate robustness, scheduling sensitivity, and downstream behavior. Comparisons with standard aggregation rules, a spectral oracle, and Heavy-Edge Matching show that performance depends on the graph family and evaluation objective. The proposed method therefore provides a theoretically grounded aggregation rule for restricted graph motifs rather than a universally optimal coarsening framework.