α-Cut Sensitivity Analysis in Fuzzy Social Networks: Change Points and Stable Threshold Intervals

Abstract

Thresholding a fuzzy social network at one unexamined membership level can produce conclu-sions that are artifacts of that threshold. This paper develops an exact a-cut sensitivity analysis for topology, community connectivity, and node ranking. Every graph statistic is shown to be a right-continuous step function whose possible change points are the distinct edge memberships. This permits finite, lossless evaluation of the entire threshold path. A multivariate change index combines component mergers, largest-component growth, clustering jumps, centralization changes, and Kendall rank turnover. An exact descending union-find algorithm detects con-nectivity transitions, while statistic-specific update rules handle triangles and degrees. Stable intervals, critical thresholds, rank ambiguity, and area-under-profile values are computed from the same finite sequence. In an eight-node fuzzy social network, the largest component grows from two to eight vertices, full connectivity appears when a falls through 0.58, and degree leadership changes repeatedly. The example demonstrates that a single alpha = 0.5 analysis would conceal the structural transition at 0.58 and the triangle-induced clustering change at 0.83.