$\alpha$-Conditional Distributivity Between Continuous t-Norms and t-Conorms With Effective Application to Suspicious Network Activity Detection

Conditional distributivity (or restricted distributivity) is a weakened form of distributivity governed by predefined constraints, serving as the theoretical cornerstone of fuzzy integral theory. Current approaches to conditional distributivity rely on the fixed constraint threshold, limiting their adaptability. To relax this limitation, this paper proposes a novel modification termed α-conditional distributivity—a parameterized hierarchical conditional distributivity equation involving continuous tnorms and t-conorms. We characterize systematically the solutions to the α-conditional distributivity equations of different types according to the idempotency of the threshold α with respect to the underlying t-norm and t-conorm as well as the attainability of α, where the obtained results are much more complicated and much richer than those for the usual conditional distributivity equations. Building upon this theoretical advancement, we enhance the ensemble method based on α-conditional distributivity equations instead of the usual distributivity equation. Experimental validation on the CICIDS2017 benchmark demonstrates that the refined framework achieves significant improvements over the original baseline across metrics including accuracy, false positive rate, precision, F1-score, Matthews correlation coefficient, and Cohen's kappa, thereby confirming its efficacy in identifying suspicious network activities.