Physical Science: New Insights and Developments Vol. 5

Network reliability is a fundamental measure for assessing the ability of a network to maintain successful connectivity under component failures. In many practical systems, such as communication networks, transportation systems, power grids, and pipeline networks, reliability evaluation depends on whether selected (possibly all) nodes remain mutually connected despite failures of edges. This chapter presents a Boolean-based framework for evaluating the all-terminal reliability of bidirectional networks. Network success and failure are formulated as coherent switching functions of independent edge success indicators, and their probabilistic expectations are obtained through the probability transform after deriving probability-ready expressions. The proposed approach integrates path-set, spanning-tree, and cut-set methods within a unified switching-algebraic setting. The all-terminal success is expressed as the conjunction of selected two-terminal successes and, equivalently, as a disjunction of spanning trees, whereas the all-terminal failure is represented through a disjunction of vertex cut-sets. Regular Karnaugh maps are used to visualise coherence, absorption, complementation, prime implicants, and disjointness, while variable-entered Karnaugh maps are used to provide compact factored representations that preserve statistical independence and facilitate the derivation of probability-ready expressions. The methodology is demonstrated for a running example of a five-node, seven-edge network. For this example, the all-terminal success function comprises 21 spanning trees and is shellable, whereas the complementary failure function comprises 10 vertex cut-sets only but is non-shellable. The case of components with equal reliabilities is further examined through derivation of reliability and unreliability polynomials, comparison with related k-out-of-n systems, and identification of the lower boundary of useful redundancy. In addition, Boolean-based structural importance measures, including frequencies of occurrence in spanning trees or vertex cut-sets, Banzhaf indices, and Shapley–Shubik indices, are evaluated. The results indicate that Boolean algebra and Karnaugh-map techniques provide a transparent and effective approach for network-reliability computation, visualization, and structural assessment.