Editorial for the Special Issue on “Fault Trees and Attack Trees: Extensions, Solution Methods, and Applications”

Fault Trees are well-known models for the reliability analysis of systems, used to compute several kinds of qualitative and quantitative measures, such as minimal cut-sets, system failure probability, sensitivity (importance) indices, etc [...]

A graphical method for breaking logical loops based on multi-tree structure

Logical loops or circular logics, interpreted as circular supporting relations among systems, remain a longstanding challenge in the probabilistic safety assessment (PSA). Logical loops are commonly found in complex industrial systems. Due to the existence of the logical loops, the minimal cut sets cannot be directly obtained. In order to solve this problem, the logical loops should be broken properly. This paper proposes a graphical method based on multi-tree structure. By constructing the simplified multi-tree, logical loops both in linearly and non-linearly interrelated systems are solved. To illustrate this method, examples of linearly interrelated systems and non-linearly interrelated systems are given in this paper. As a supplement, this method is applied to the well-known complex logical loops in the nuclear power plant. It shows that this method is highly intuitive and efficient by means of graphs.

Speeding up the core algorithm for the dual calculation of minimal cut sets in large metabolic networks

Background The concept of minimal cut sets (MCS) has become an important mathematical framework for analyzing and (re)designing metabolic networks. However, the calculation of MCS in genome-scale metabolic models is a complex computational problem. The development of duality-based algorithms in the last years allowed the enumeration of thousands of MCS in genome-scale networks by solving mixed-integer linear problems (MILP). A recent advancement in this field was the introduction of the MCS2 approach. In contrast to the Farkas-lemma-based dual system used in earlier studies, the MCS2 approach employs a more condensed representation of the dual system based on the nullspace of the stoichiometric matrix, which, due to its reduced dimension, holds promise to further enhance MCS computations. Results In this work, we introduce several new variants and modifications of duality-based MCS algorithms and benchmark their effects on the overall performance. As one major result, we generalize the original MCS2 approach (which was limited to blocking the operation of certain target reactions) to the most general case of MCS computations with arbitrary target and desired regions. Building upon these developments, we introduce a new MILP variant which allows maximal flexibility in the formulation of MCS problems and fully leverages the reduced size of the nullspace-based dual system. With a comprehensive set of benchmarks, we show that the MILP with the nullspace-based dual system outperforms the MILP with the Farkas-lemma-based dual system speeding up MCS computation with an averaged factor of approximately 2.5. We furthermore present several simplifications in the formulation of constraints, mainly related to binary variables, which further enhance the performance of MCS-related MILP. However, the benchmarks also reveal that some highly condensed formulations of constraints, especially on reversible reactions, may lead to worse behavior when compared to variants with a larger number of (more explicit) constraints and involved variables. Conclusions Our results further enhance the algorithmic toolbox for MCS calculations and are of general importance for theoretical developments as well as for practical applications of the MCS framework.

Binomial edge ideals of generalized block graphs

We classify generalized block graphs whose binomial edge ideals admit a unique extremal Betti number. We prove that the Castelnuovo–Mumford regularity of binomial edge ideals of generalized block graphs is bounded below by [Formula: see text], where [Formula: see text] is the number of minimal cut sets of the graph [Formula: see text] and obtain an improved upper bound for the regularity in terms of the number of maximal cliques and pendant vertices of [Formula: see text].

Enumeration of the minimal node cutsets based on necessary minimal paths

<p>Reliability evaluation is an important research field for a complex network. The most popular methods for such evaluation often use Minimal Cuts (MC) or Minimal paths (MP). Nonetheless, few algorithms address the issue of the enumeration of all minimal cut sets from the source node s to the terminal node t when only the nodes of the network are subject to random failures. This paper presents an effective algorithm which enumerates all minimal node cuts sets of a network. The proposed algorithm runs in two steps: The first one is used to generate a subset of paths, called necessary minimal paths, instead of all minimal paths. Whereas, the second step stands to build all minimal cutsets from the necessary minimal paths.</p>