Corrigendum to "Degree-Based Approximations for Network Reliability Polynomials". Comment on J. Complex Networks 2025, 13, cnaf001

Abstract

Our original paper \cite{VanMieghem2025} described the stochastic approximation ${\overline{rel}}_G(p)=[1-\ddot{I}{•}_D(1-p){]}^N$ in \cite[eq. (2.2)]{VanMieghem2025} and the first-order approximation $(R_1{)}_G(p)={\prod }_{i=1}^N[1-(1-p{)}^{d_i}]$ in \cite[eq. (4.1)]{VanMieghem2025} as upper bounds for the all-terminal reliability polynomial \(rel_G(p)\). The present corrigendum clarifies that the unique upper bound is \(\Pr[\hat D_{\min}\geq 1]\), which is difficult to compute exactly, because we must account for correlated node-isolation events. Both the stochastic approximation ${\overline{rel}}_G$ and the first-order approximation $(R_1{)}_G$ ignore those correlations, assume independence and, consequently, do not always upperbound \(rel_G(p)\) as stated previously. The complete graph \(K_{3}\) is a counterexample, where both approximations lie below the exact reliability polynomial $re{l}_{K_3}(p)$, illustrating that they are not upper bounds. Moreover, as claimed in \cite{VanMieghem2025}, the first-order approximation $(R_1{)}_G$ is not always more accurate than the stochastic approximation ${\overline{rel}}_G$. We show by an example that the relative accuracy of the stochastic approximation ${\overline{rel}}_G$ and the first-order approximation $(R_1{)}_G$ varies with the graph $G$ and the link operational probability $p$. }{network robustness, node failure, probabilistic graph, reliability polynomial