Exact two-terminal reliability of some directed networks

The calculation of network reliability in a probabilistic context has long been an issue of practical and academic importance. Conventional approaches (determination of bounds, sums of disjoint products algorithms, Monte Carlo evaluations, studies of the reliability polynomials, etc.) only provide approximations when the network's size increases, even when nodes do not fail and all edges have the same reliability p. We consider here a directed, generic graph of arbitrary size mimicking real-life long-haul communication networks, and give the exact, analytical solution for the two-terminal reliability. This solution involves a product of transfer matrices, in which individual reliabilities of edges and nodes are taken into account. The special case of identical edge and node reliabilities (p and rho, respectively) is addressed. We consider a case study based on a commonly-used configuration, and assess the influence of the edges being directed (or not) on various measures of network performance. While the two-terminal reliability, the failure frequency and the failure rate of the connection are quite similar, the locations of complex zeros of the two-terminal reliability polynomials exhibit strong differences, and various structure transitions at specific values of rho. The present work could be extended to provide a catalog of exactly solvable networks in terms of reliability, which could be useful as building blocks for new and improved bounds, as well as benchmarks, in the general case

Exact solutions for the two- and all-terminal reliabilities of the Brecht-Colbourn ladder and the generalized fan

The two- and all-terminal reliabilities of the Brecht-Colbourn ladder and the generalized fan have been calculated exactly for arbitrary size as well as arbitrary individual edge and node reliabilities, using transfer matrices of dimension four at most. While the all-terminal reliabilities of these graphs are identical, the special case of identical edge ($p$) and node ($\rho$) reliabilities shows that their two-terminal reliabilities are quite distinct, as demonstrated by their generating functions and the locations of the zeros of the reliability polynomials, which undergo structural transitions at $\rho = \displaystyle {1/2}$

A contribution to the evaluation and optimization of networks reliability

L’évaluation de la fiabilité des réseaux est un problème combinatoire très complexe qui nécessite des moyens de calcul très puissants. Plusieurs méthodes ont été proposées dans la littérature pour apporter des solutions. Certaines ont été programmées dont notamment les méthodes d’énumération des ensembles minimaux et la factorisation, et d’autres sont restées à l’état de simples théories. Cette thèse traite le cas de l’évaluation et l’optimisation de la fiabilité des réseaux. Plusieurs problèmes ont été abordés dont notamment la mise au point d’une méthodologie pour la modélisation des réseaux en vue de l’évaluation de leur fiabilités. Cette méthodologie a été validée dans le cadre d’un réseau de radio communication étendu implanté récemment pour couvrir les besoins de toute la province québécoise. Plusieurs algorithmes ont aussi été établis pour générer les chemins et les coupes minimales pour un réseau donné. La génération des chemins et des coupes constitue une contribution importante dans le processus d’évaluation et d’optimisation de la fiabilité. Ces algorithmes ont permis de traiter de manière rapide et efficace plusieurs réseaux tests ainsi que le réseau de radio communication provincial. Ils ont été par la suite exploités pour évaluer la fiabilité grâce à une méthode basée sur les diagrammes de décision binaire. Plusieurs contributions théoriques ont aussi permis de mettre en place une solution exacte de la fiabilité des réseaux stochastiques imparfaits dans le cadre des méthodes de factorisation. A partir de cette recherche plusieurs outils ont été programmés pour évaluer et optimiser la fiabilité des réseaux. Les résultats obtenus montrent clairement un gain significatif en temps d’exécution et en espace de mémoire utilisé par rapport à beaucoup d’autres implémentations. Mots-clés: Fiabilité, réseaux, optimisation, diagrammes de décision binaire, ensembles des chemins et coupes minimales, algorithmes, indicateur de Birnbaum, systèmes de radio télécommunication, programmes.Efficient computation of systems reliability is required in many sensitive networks. Despite the increased efficiency of computers and the proliferation of algorithms, the problem of finding good and quickly solutions in the case of large systems remains open. Recently, efficient computation techniques have been recognized as significant advances to solve the problem during a reasonable period of time. However, they are applicable to a special category of networks and more efforts still necessary to generalize a unified method giving exact solution. Assessing the reliability of networks is a very complex combinatorial problem which requires powerful computing resources. Several methods have been proposed in the literature. Some have been implemented including minimal sets enumeration and factoring methods, and others remained as simple theories. This thesis treats the case of networks reliability evaluation and optimization. Several issues were discussed including the development of a methodology for modeling networks and evaluating their reliabilities. This methodology was validated as part of a radio communication network project. In this work, some algorithms have been developed to generate minimal paths and cuts for a given network. The generation of paths and cuts is an important contribution in the process of networks reliability and optimization. These algorithms have been subsequently used to assess reliability by a method based on binary decision diagrams. Several theoretical contributions have been proposed and helped to establish an exact solution of the stochastic networks reliability in which edges and nodes are subject to failure using factoring decomposition theorem. From this research activity, several tools have been implemented and results clearly show a significant gain in time execution and memory space used by comparison to many other implementations. Key-words: Reliability, Networks, optimization, binary decision diagrams, minimal paths set and cuts set, algorithms, Birnbaum performance index, Networks, radio-telecommunication systems, programs

Evaluation of Two Terminal Reliability of Fault-tolerant Multistage Interconnection Networks

This paper iOntroduces a new method based on multi-decomposition for predicting the two terminal reliability of fault-tolerant multistage interconnection networks. The method is well supported by an efficient algorithm which runs polynomially. The method is well illustrated by taking a network consists of eight nodes and twelve links as an example. The proposed method is found to be simple, general and efficient and thus is as such applicable to all types of fault-tolerant multistage interconnection networks. The results show this method provides a greater accurate probability when applied on fault-tolerant multistage interconnection networks. Reliability of two important MINs are evaluated by using the proposed method

A memory efficient algorithm for network reliability

We combine the Augmented Ordered Binary Decision Diagram (OBDD-A) with the use of boundary sets to create a method for computing the exact K-terminal or all-terminal reliability of an undirected network with failed edges and perfect vertices. We present the results of implementing this algorithm and show that the execution time is comparable with the state of the art and the space requirement is greatly reduced. Indeed the space remains constant when networks increase in size but maintain their structure and maximum boundary set size; with the same amount of memory used for computing a 312 and a 31000 grid network

An efficient algorithm for computing exact system and survival signatures of K-terminal network reliability

An efficient algorithm is presented for computing exact system and survival signatures of K-terminal reliability in undirected networks with unreliable edges. K-terminal reliability is defined as the probability that a subset K of the network nodes can communicate with each other. Signatures have several advantages over direct reliability calculation such as enabling certain stochastic comparisons of reliability between competing network topology designs, extremely fast repeat computation of network reliability for different edge reliabilities and computation of network reliability when failures of edges are exchangeable but not independent. Existing methods for computation of signatures for K-terminal network reliability require derivation of cut-sets or path-sets which is only feasible for small networks due to the computational expense. The new algorithm utilises binary decision diagrams, boundary set partition sets and simple array operations to efficiently compute signatures through a factorisation of the network edges. The performance and advantages of the algorithm are demonstrated through application to a set of benchmark networks and a sensor network from an underground mine

k-connectivity of Random Graphs and Random Geometric Graphs in Node Fault Model

k-connectivity of random graphs is a fundamental property indicating reliability of multi-hop wireless sensor networks (WSN). WSNs comprising of sensor nodes with limited power resources are modeled by random graphs with unreliable nodes, which is known as the node fault model. In this paper, we investigate k-connectivity of random graphs in the node fault model by evaluating the network breakdown probability, i.e., the disconnectivity probability of random graphs after stochastic node removals. Using the notion of a strongly typical set, we obtain universal asymptotic upper and lower bounds of the network breakdown probability. The bounds are applicable both to random graphs and to random geometric graphs. We then consider three representative random graph ensembles: the Erdos-Renyi random graph as the simplest case, the random intersection graph for WSNs with random key predistribution schemes, and the random geometric graph as a model of WSNs generated by random sensor node deployment. The bounds unveil the existence of the phase transition of the network breakdown probability for those ensembles.Comment: 6 page

Inferring gene ontologies from pairwise similarity data.

MotivationWhile the manually curated Gene Ontology (GO) is widely used, inferring a GO directly from -omics data is a compelling new problem. Recognizing that ontologies are a directed acyclic graph (DAG) of terms and hierarchical relations, algorithms are needed that: analyze a full matrix of gene-gene pairwise similarities from -omics data; infer true hierarchical structure in these data rather than enforcing hierarchy as a computational artifact; and respect biological pleiotropy, by which a term in the hierarchy can relate to multiple higher level terms. Methods addressing these requirements are just beginning to emerge-none has been evaluated for GO inference.MethodsWe consider two algorithms [Clique Extracted Ontology (CliXO), LocalFitness] that uniquely satisfy these requirements, compared with methods including standard clustering. CliXO is a new approach that finds maximal cliques in a network induced by progressive thresholding of a similarity matrix. We evaluate each method's ability to reconstruct the GO biological process ontology from a similarity matrix based on (a) semantic similarities for GO itself or (b) three -omics datasets for yeast.ResultsFor task (a) using semantic similarity, CliXO accurately reconstructs GO (>99% precision, recall) and outperforms other approaches (<20% precision, <20% recall). For task (b) using -omics data, CliXO outperforms other methods using two -omics datasets and achieves ∼30% precision and recall using YeastNet v3, similar to an earlier approach (Network Extracted Ontology) and better than LocalFitness or standard clustering (20-25% precision, recall).ConclusionThis study provides algorithmic foundation for building gene ontologies by capturing hierarchical and pleiotropic structure embedded in biomolecular data

International Competition on Graph Counting Algorithms 2023

This paper reports on the details of the International Competition on Graph Counting Algorithms (ICGCA) held in 2023. The graph counting problem is to count the subgraphs satisfying specified constraints on a given graph. The problem belongs to #P-complete, a computationally tough class. Since many essential systems in modern society, e.g., infrastructure networks, are often represented as graphs, graph counting algorithms are a key technology to efficiently scan all the subgraphs representing the feasible states of the system. In the ICGCA, contestants were asked to count the paths on a graph under a length constraint. The benchmark set included 150 challenging instances, emphasizing graphs resembling infrastructure networks. Eleven solvers were submitted and ranked by the number of benchmarks correctly solved within a time limit. The winning solver, TLDC, was designed based on three fundamental approaches: backtracking search, dynamic programming, and model counting or #SAT (a counting version of Boolean satisfiability). Detailed analyses show that each approach has its own strengths, and one approach is unlikely to dominate the others. The codes and papers of the participating solvers are available: https://afsa.jp/icgca/.Comment: https://afsa.jp/icgca