arXiv:2607.01546v2 Announce Type: replace Abstract: Rankings derived from weighted scoring functions are widely used in university rankings and employment candidate evaluations. Since ranking weights are often chosen by organizations or analysts, ranking stability asks whether a reported ranking persists under reasonable weight changes. Prior work defines volume-based stability as the fraction of the weight space that induces the target ranking exactly. This exact-match requirement can be too blunt: once a perturbed weight vector produces a different ranking, exact stability gives it no credit, whether the change replaces the top-ranked item or only swaps two nearly tied lower-ranked items. We propose general stability, a distance-based generalization that aggregates ranking regions according to a user-defined distance from the target ranking. This lets users specify which ranking changes matter in the application, while recovering exact stability as a special case. We extend existing methods and identify distance properties that permit faster general-stability computation with accuracy guarantees. Our sampling algorithm, $GSC_{md}$, can take exponential time in the worst case, but a property we call uniform boundedness gives a sample budget independent of database size and dimension at fixed accuracy and confidence. We develop Prefix-GSC, a sampling optimization that reduces sorting for distances satisfying a further property, tail decay with bounded influence. We also develop Conv-SC to approximate stability through convex volumes for distances satisfying a property we term quasiconvexity. Experiments on seven real-data sources and generated inputs show how distance affects robustness conclusions and when the algorithms save time. On three 500-item university datasets, Prefix-GSC achieves median query speedups of $1.22$--$1.98\times$ over $GSC_{md}$ at 2% relative error and 95% confidence, including method setup.