The Unified Algebraic-Behavioral Family Stability Index (UFSI): A Computational Simulation Study Using Synthetic Data — with Formal Mathematical Properties and Machine Learning Comparison (Al-Farooq Journal of Sciences, 2026)
Family dissolution represents a growing socio-economic challenge in Libya. Current diagnostic tools lack the quantitative precision and dynamic predictive capacity necessary for pre-crisis intervention.This study formulates and validates the mathematical properties of the Unified Family Stability Index (UFSI), a hybrid model integrating social role theory with modified Perron-Frobenius algebra and a Decision Asymmetry Index.We employed a three-stage methodological design: (1) formal proof of mathematical properties (existence, continuity, monotonicity); (2) Monte Carlo simulation (N=10,000) using a Weighted Influence Matrix with 10-fold cross-validation; (3) performance comparison against ML models using DeLong Test and Calibration Analysis.Simulation demonstrated high classification efficiency (AUC = 0.94; 95% CI: 0.92–0.96), with a statistically significant advantage over XGBoost (p < 0.001). A preliminary operational threshold was identified at UFSI ≈ 71.2.UFSI offers a coherent, interpretable framework, pending empirical calibration for external validity.
Keywords: Family Stability; Mathematical Modeling; Monte Carlo Simulation; Spectral Radius; Decision Asymmetry; Computational Social Science; Libya; UFSI