Uncertainty-Driven Stability Analysis of Minimum Spanning Tree Under Multiple Risk Variations

The Minimum Spanning Tree (MST) problem addresses the challenge of identifying optimal network pathways for critical infrastructure systems, including transportation grids, communication backbones, power distribution networks, and reliability optimization frameworks. However, inherent uncertainties stemming from disruptive events demand robust analytical models for effective decision-making. This research introduces an uncertainty-theoretic framework to assess MST stability in uncertain network environments through novel constructs: lower set tolerance (LST) and dual lower set tolerance (DLST). Both LST and DLST provide quantifiable measures characterizing the resilience of element sets relative to edge-weighted MST configurations. LST captures the maximum simultaneous risk variation preserving current MST optimality, while DLST identifies the minimal variation required to invalidate it. We evaluate MST robustness by integrating uncertain reliability measures and risk factors, with emphasis on computational methods for set tolerance determination. To overcome computational hurdles in set tolerance derivation, we establish bounds and exact formulations within an uncertainty programming paradigm, offering enhanced efficiency compared with conventional re-optimization techniques.