Fuzzifier-Induced Robustness Phase Transitions in Median-Centred Fuzzy Block Aggregation

Abstract

Fuzzifier-Induced Robustness Phase Transitions in Median-Centred Fuzzy Block Aggregation develops and tests a robust fuzzy aggregation framework for data divided into blocks, where some blocks may be contaminated, extreme, or otherwise unreliable. The central problem is that fuzzy membership weighting does not automatically guarantee robustness. Depending on how rapidly the membership assigned to a distant block decays, an extreme contaminated block may have vanishing influence, bounded influence, or unbounded influence on the final aggregate. The framework provides a mathematical characterisation of these regimes through a tail-leverage criterion. For the standard noise-cluster membership law, the analysis yields a sharp robustness phase transition governed by the fuzzifier and aggregation power. On the robust side of the transition, sub-majority block contamination remains bounded; beyond the transition, a single sufficiently extreme block can cause breakdown. The framework also establishes a fuzzy-to-crisp limit in which graded memberships converge to hard residual trimming, together with finite-sample replacement-breakdown results, adaptive-scale extensions, vector-valued aggregation, and finite-variance concentration guarantees. The archive contains the complete computational material used for the numerical study: make_figures.py: generates all numerical experiments and publication figures. regression_tests.py: runs the numerical regression and edge-case tests used to check the theoretical results computationally. numerical_summary.txt: records the principal numerical outputs from the experiments. Six figure files in PDF format corresponding to the numerical illustrations. requirements.txt: lists the Python package dependencies required to run the code. environment.txt: records the software environment used for the archived computations. A regression report recording the outcome of the automated numerical checks. No external dataset is required. All numerical data used in the experiments are generated synthetically by the supplied code. The software is provided as research code for transparency and reproducibility. It is released as-is and will not be maintained, updated, extended, or supported. Compatibility with future versions of Python or its dependencies is not guaranteed.