Laplacian Ranking (LaRa): A Convex Graph-Regularized Model for Multi-Criteria Ranking

Abstract

Laplacian Ranking (LaRa) is a graph-regularized multi-criteria decision analysis method that incorporates relational geometry among alternatives into ranking while preserving convexity, interpretability and deterministic reproducibility. LaRa combines quantile normalization, metric-weighted distances, a sparse similarity graph, a reference-oriented prior and an optional quadratic pairwise target-gap term. Its base score is the unique minimizer of a strictly convex quadratic objective, obtained by solving a symmetric positive definite linear system, with an optional deterministic dominance-repair step. We evaluate LaRa on seven literature-grounded real-data problems spanning materials, manufacturing equipment, industrial robotics, renewable-energy siting, and energy-technology selection, together with a controlled synthetic pharmaceutical-style benchmark. Five cross-domain cases use one common fixed configuration and sensitivity grid rather than dataset-specific tuning. Results show that the graph contribution is regime-dependent. It is ordinally inert for cryogenic materials and clean-energy technologies, measurable but non-disruptive for flywheel selection, improves concordance with published rankings for CNC machine tools, arc-welding robots and Colombian solar-PV siting, and reduces concordance for gear-material selection, which is retained as an explicit failure boundary. On the fixed synthetic benchmark, the high-smoothing configuration attains Kendall's ${\tau }_b=0.987$ and Spearman's $\rho =0.998$, reduces mean absolute rank error from 0.556 for TOPSIS to 0.111, and exactly recovers the Top-3 set. These findings position LaRa as a relational refinement of pointwise MCDA whose usefulness depends on whether neighbourhood geometry is substantively meaningful and empirically supported. Accordingly, graph smoothing should be activated as a conditional modelling component and accompanied by sensitivity analysis rather than assumed beneficial in practical MCDA applications.