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

TL;DR

The corrigendum clarifies that the two degree-based approximations to the all-terminal reliability polynomial, $\overline{\mathrm{rel}}_G(p)$ and $(R_1)_G(p)$, approximate the correlated event $\Pr[\hat{D}_{\min}\ge 1]$ under independence rather than providing universal upper bounds for $\mathrm{rel}_G(p)$. It shows that $\mathrm{rel}_G(p)\le \Pr[\hat{D}_{\min}\ge 1]$ but neither approximation universally upper bounds $\mathrm{rel}_G(p)$, as node-isolation correlations can violate the independence assumption. The authors present counterexamples, including $K_3$, where both approximations lie below the true reliability, and a modified circulant graph where the stochastic approximation can be more accurate than the first-order one for a broad range of $p$, thereby refuting a universal ordering between the two. Overall, the work emphasizes that accuracy depends on graph structure (notably degree correlations/assortativity) and the operating probability $p$, and that there is no universal hierarchy between these degree-based bounds and the reliability polynomial.

Abstract

Our original paper \cite{VanMieghem2025} described the stochastic approximation $\overline{rel}_G(p)=\bigl[1-φ_D(1-p)\bigr]^{N}$ in \cite[eq. (2.2)]{VanMieghem2025} and the first-order approximation $(R_1)_G(p)=\prod_{i=1}^{N}\!\bigl[1-(1-p)^{d_i}\bigr]$ 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 $rel_{K_3}(p)$, 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

Figures (2)

• Figure 1: Illustration of dependence between events {Node $i$ is isolated} and {Node $j$ is isolated}. Each link is independently operational with probability $p$. The probability of event {Node $i$ is isolated} is $\Pr[\text{Node $i$ is isolated}]=1-p^4$. For node $j$, the probability is $\Pr[\text{Node $j$ is isolated}]=1-p^3$. The joint probability is $\Pr[\{\text{Node $i$ is isolated}\}\cap \{\text{Node $j$ is isolated}\}]=1-p^6$. If the two events were independent, the probability is $\Pr[\{\text{Node $i$ is isolated}\}\cap \{\text{Node $j$ is isolated}\}]=\Pr[\{\text{Node $i$ is isolated}\}]\Pr[\{\text{Node $j$ is isolated}\}]=1-p^3-p^4+p^7$.
• Figure 2: The stochastic approximation, first-order approximation and Monte Carlo simulations for a modified circulant on 15 nodes (nodes numbered $1$–$15$): start from the complete graph $K_{15}$ and delete the nine links $1\!-\!2,\,1\!-\!3,\,1\!-\!4,\,1\!-\!6,\,1\!-\!7,\,1\!-\!8,\,1\!-\!13,\,1\!-\!14,\,1\!-\!15$. In the resulting graph, node $1$ has a degree of 5 (node $1$ is linked only to nodes $5,9,10,11,12$), while all other nodes have degree $13$ or $14$. Properties of circulant matrices of small-world graphs are deduced in PVM_graphspectra_second_edition.