All-terminal Network Reliability Research Articles

A deep neural network and Bayesian method based framework for all-terminal network reliability estimation considering degradation

Most research on network reliability has considered links to have a binary state, i.e., functioning or failed, whereas nodes are considered flawless. In a more realistic scenario, both links and nodes might fail or may exhibit degradation behavior before failing. This study develops a framework to estimate the all-terminal reliability of a network that considers the degradation and probability of failure of all nodes and links in a network. Unlike previous works on network reliability that considered constant reliability for links, this paper considers the reliability of links, nodes, and the network as functions of time. In the proposed framework, the Bayesian methods (BM) are employed to estimate the reliability of links and nodes as functions of time considering degradation data. Due to the complexity of the all-terminal reliability problem, and to get fast estimations of the reliability of a network, an integration of Monte Carlo (MC) and Deep Neural Networks (DNNs) is proposed. The proposed MC algorithm can estimate the network reliability for given nodes and links reliability values. To speed up the calculation, a DNN model is integrated into the framework, thus enabling accurate and fast estimation of network reliability for given link and node reliability values. The DNN accuracy, based on the RMSE (0.01460), outperforms previous traditional artificial neural network (ANN) approaches. Moreover, the DNN model takes 0.3 ms to compute the reliability for any given links and reliability values. The proposed framework can provide not only reliability point estimates but also credible intervals. Finally, we take advantage of Bayesian methods to integrate new data into the framework as they become available. The framework uses the new data to refine and further update the degradation model parameters and the prediction of the reliability of links, nodes, and the network. The proposed methodology has been demonstrated with the real-world network topology Ion (125 nodes, 150 links) with actual degradation data.

Estimation of all-terminal network reliability using press forward GA with stochastic selection method

Network reliability is an important measure of how well the network meets its design aim and mathematically is the probability that a network will perform communication efficiently for at least a given period of time. The reliability analysis is used for determining the probability of correct functioning of a system in terrain conditions. Estimating the all-terminal network reliability is an NP-hard problem, the press forward Genetic Algorithm is also an alternative to solve the problem of all-terminal network reliability. This paper presents a reliability evaluation method by using a feed-forward genetic algorithm with a random selection of nodes and links i.e., estimate the network reliability such that the flow is not less than a given threshold. To verify the efficiency of an algorithm, various sets of runs are applied to identify the most reliable network and the algorithm is tested on the network size of 2,4,6,8, --------256 nodes with link N(N-1)/2. The results of the algorithm are obtained for all-terminal reliability problem based on close analysis of the complexity of the network. Our assumption for reliability evaluation of node 1 to node N, with link reliability of 50%. Another result of the proposed GA approach is the significant reduction in computational time.

All-terminal network reliability estimation using convolutional neural networks

Estimating the all-terminal network reliability by using artificial neural networks (ANNs) has emerged as a promissory alternative to classical exact NP-hard algorithms. Approaches based on traditional ANNs have usually considered the network reliability upper bound as part of the inputs, which implies additional time-consuming calculations during both training and testing phases. This paper proposes the use of Convolutional Neural Networks (CNNs), without the reliability upper-bound as an input, to address the all-terminal network reliability estimation problem. The present study introduces a multidimensional matrix format to embed the topological and link reliability information of networks. The unique contribution of this article is the method to capture the topology of a network in terms of its adjacency matrix, link reliability, and topological attributes providing a novel use of CNN beyond image classification. Since CNNs have been successful for image classification, appropriate modifications are needed and introduced to use them in the estimation of network reliability. A regression output layer is proposed, preceded by a sigmoid layer to achieve predictions within the range of reliability characteristic, a feature that some previous ANN-based works lack. Several training parameters together with a filter multiplier (CNN architecture parameter) were investigated. The actual values and the ones predicted with the best trained CNN were compared in the light of RMSE (0.04406) and p-value (0.3) showing non-significant difference. This study provides evidence supporting the hypothesis that the network reliability can be estimated by CNNs from its topology and link reliability information, embedded as an image-like multidimensional matrix. Another important result of the proposed approach is the significant reduction in computational time. An average of 1.18 ms/network was achieved by the CNN, whereas backtracking exact algorithm took around 500 s/network.

A Polynomial-Time Approximation Algorithm for All-Terminal Network Reliability

We give a fully polynomial-time randomized approximation scheme (FPRAS) for the all-terminal network reliability problem, which is to determine the probability that, in a undirected graph, assuming each edge fails independently, the remaining graph is still connected. Our main contribution is to confirm a conjecture by Gorodezky and Pak (Random Struct. Algorithms, 2014), that the expected running time of the "cluster-popping" algorithm in bi-directed graphs is bounded by a polynomial in the size of the input.

Quantum-inspired Boolean states for bounding engineering network reliability assessment

Significant methodological progress has taken place to quantify the reliability of networked systems over the past decades. Both numerical and analytical methods have enjoyed improvements via a host of advanced Monte Carlo simulation strategies, state space partition methods, statistical learning, and Boolean functions among others. The latter approach exploits logic to approximate network reliability assessments efficiently while offering theoretical error guarantees. In parallel, physicists have made progress modeling complex systems via tensor networks (TNs), particularly quantum many-body systems. Inspired by the representation power of quantum TNs, this paper offers a new approach to efficiently bound all-terminal network reliability (ATR). It does so by exactly solving a related network Boolean satisfiability counting problem (or #SATNET), represented as a TN, by counting configurations in which all network nodes are connected to at least a neighbor. Our #SATNET counting outperforms state-of-the-art approximate and exact counters for the same problem as shown for challenging two-dimensional lattice networks of increasing size. While the over-counting from #SATNET increases exponentially relative to the number of configurations that satisfy ATR problems (or #RELAT), the bias is predictable for ideal networks, such as lattices, and our upper bound is guaranteed with 100% confidence. This bound also cuts through the upper bound from other analytical methods for the ATR problem, such as recursive decomposition algorithms (RDA)—a desirable feature when exact or approximate methods with error guarantees fail to scale. Clearly, our goal is not to solve the general stochastic network reliability problem, which remains a #P problem in the computational complexity hierarchy (i.e., a counting version of non-deterministic polynomial time [NP] problems for which there is no known polynomial time algorithms to find their solutions). Instead, we present a novel bounding technique for network reliability, which relies on exactly counting configurations for our network satisfiability problem by using quantum computing principles. We offer a proof for the counting bound to hold in a connectivity ATR sense, and illustrate trends with cubic and lattice networks. Overall, the proposed method provides an alternative to available system reliability assessment approaches, and opens directions for future research, especially as discoveries in logic, algebraic projections, and quantum computation continue to accrue.

Improvements in the Efficiency of Cumulative Updating of All-Terminal Network Reliability

This paper contains some algorithms and recommendations that enable increasing efficiency of cumulative updating of all-terminal reliability for a network with unreliable links. The existence of cutnodes, 2-node cuts, and chains in a network structure can be used for faster calculations.

Chip firing and all-terminal network reliability bounds

The (all-terminal) reliability of a graph G is the probability that all vertices are in the same connected component, given that vertices are always operational but edges fail independently each with probability p . Computing reliability is #P-complete, and hence is expected to be intractable. Consequently techniques for efficiently (and effectively) bounding reliability have been the major thrust of research in the area. We utilize a deep connection between reliability and chip firings on graphs to improve previous bounds for reliability.

All-Terminal Network Reliability Using Recursive Truncation Algorithm

Exact calculation of all-terminal network reliability is a hard problem; its computational complexity grows exponentially with the number of nodes and links in the network. We propose the Recursive Truncation Algorithm (RTA), a bounding approximation algorithm, to estimate the all-terminal reliability of a given network with a pre-specified accuracy. RTA scans all minimal cutsets of the graph representing the network, and finds the weak cutsets of the graph by comparing failure probabilities of cutsets to an adaptive threshold which depends on the approximation accuracy. We calculate the unreliability of the network versus the probabilities of occurrence of failure in the weak cutsets, and the probabilities of co-occurrence of failure in several weak cutsets, simultaneously. In addition to the all-terminal reliability, the RTA computes an upper, and a lower bound for the estimated reliability. We demonstrate that the estimated all-terminal reliability of a given network is within a pre-specified accuracy, and is more precise than those obtained by existing methods.

All-terminal network reliability optimization via probabilistic solution discovery

This paper presents a new algorithm that can be readily applied to solve the all-terminal network reliability allocation problems. The optimization problem solved considers the minimization of the network design cost subject to a known constraint on all-terminal reliability by assuming that the network contains a known number of functionally equivalent components (with different performance specifications) that can be used to provide redundancy. The algorithm is based on two major steps that use a probabilistic solution discovery approach and Monte Carlo simulation to generate the quasi-optimal network designs. Examples for different sizes of all-terminal networks are used throughout the paper to illustrate the approach. The results obtained for the larger networks with unknown optima show that the quality of the solutions generated by the proposed algorithm is significantly higher with respect to other approaches and that these solutions are obtained from restricted solution search space. Although developed for all-terminal reliability optimization, the algorithm can be easily applied in other resource-constrained allocation problems.

Estimation of all-terminal network reliability using an artificial neural network

The exact calculation of all-terminal network reliability is an NP-hard problem, with computational effort growing exponentially with the number of nodes and links in the network. During optimal network design, a huge number of candidate topologies are typically examined with each requiring a network reliability calculation. Because of the impracticality of calculating all-terminal network reliability for networks of moderate to large size, Monte Carlo simulation methods to estimate network reliability and upper and lower bounds to bound reliability have been used as alternatives. This paper puts forth another alternative to the estimation of all-terminal network reliability — that of artificial neural network (ANN) predictive models. Neural networks are constructed, trained and validated using the network topologies, the link reliabilities, and a network reliability upperbound as inputs and the exact network reliability as the target. A hierarchical approach is used: a general neural network screens all network topologies for reliability followed by a specialized neural network for highly reliable network designs. Both networks with identical link reliability and networks with varying link reliability are studied. Results, using a grouped cross-validation approach, show that the ANN approach yields more precise estimates than the upperbound, especially in the worst cases. Using the reliability estimation methods of the ANN, the upperbound and backtracking, optimal network design by simulated annealing is considered. Results show that the ANN regularly produces superior network designs at a reasonable computational cost. Scope and purpose An important application area of operations research is the design of structures, products or systems where both technical and business aspects must be considered. One expanding design domain is the design of computer or communications networks. While cost is a prime consideration, reliability is equally important. A common reliability measure is all-terminal reliability, the probability that all nodes (computers or terminals) on the network can communicate with all others. Exact calculation of all-terminal reliability is an NP-hard problem, precluding its use during optimal network topology design, where this calculation must be made thousands or millions of times. This paper presents a novel computationally practical method for estimating all-terminal network reliability. Is shown how a neural network can be used to estimate all-terminal network reliability by using the network topology, the link reliabilities and an upperbound on all-terminal network reliability as inputs. The neural network is trained and validated on a very minute fraction of possible network topologies, and once trained, it can be used without restriction during network design for a topology of a fixed number of nodes. The trained neural network is extremely fast computationally and can accommodate a variety of network design problems. The neural network approach, an upper bound approach and an exact backtracking calculation are compared for network design using simulated annealing for optimization and show that the neural network approach yields superior designs at manageable computational cost.

Spanning tree approach in all-terminal network reliability expansion

A method is presented to maximize the reliability improvement of a communication network through a new edge addition between an existing node pair of the network. The method does not require the numerical reliability functions for determining such a node pair. The node pairs are compared by the number of spanning trees generated as a result of the connection of an additional link, and then the node pair connection, which results in generating the largest number of spanning trees, is determined. The counting process for the number of spanning trees of the expanded network is based on the degree matrix (DM) of the network.

A neural approach to topological optimization of communication networks, with reliability constraints

Consider network topological optimization under a reliability constraint. The objective is to find the topological layout of links, at minimal cost, under the constraint: all-terminal network reliability is not less than a given level of system reliability. The all-terminal reliability is Pr{every pair of nodes in the network can communicate with each other}. This paper presents a new approach based on artificial neural networks (ANN) for solving the problem. The problem is mapped onto an optimization ANN (OPTI-net) by constructing an energy function whose minimization process drives the neural network into one of its stable states. This stable state corresponds to a solution for the network design problem. The OPTI-net favors states that correspond to a selection of links with an overall reliability greater than or equal to a threshold value. Among these states it also favors the one which has the lowest total cost. Hysteresis McCulloch-Pitts neuron model is used in the solution, due to its performance and fast convergence. Considering the NP-hard complexity of the exact reliability calculation, together with the iterative behavior of the neural networks, bounds for the all-terminal reliability are used. This paper introduces new upper and lower bounds that are functions of the link selection and uses them to represent the network reliability. The neural network is tested via computer simulation using three problem sets. The first two sets are used to compare the results obtained by this method to those obtained by previous heuristics. The third test set contains five networks of larger sizes for which no results have been reported by previous methods. This paper rinds the optimal or near-optimal solutions for most of the problems in a relatively short time. The OPTI-net found many good solutions for a 50-vertex 1225-arc network in 1/2 CPU hour. For each problem instance, many solutions are found at each run of the simulator. The strengths of this neural network approach are very slowly increasing computation time with respect to network size, effective optimization, and flexibility. The OPTI-net is very effective in identifying optimal, or suboptimal, solutions even in search spaces up to /spl ap/10/sup 16/ for a fully connected network with 50 vertexes. The OPTI-net is the first approach to be applied on such large networks. The simulation results show that the neural approach is more efficient in designing networks of large sizes compared to other heuristic techniques.

A Randomized Fully Polynomial Time Approximation Scheme for the All-Terminal Network Reliability Problem

The classic all-terminal network reliability problem posits a graph, each of whose edges fails independently with some given probability. The goal is to determine the probability that the network becomes disconnected due to edge failures. This problem has obvious applications in the design of communication networks. Since the problem is ${\sharp {\cal P}}$-complete and thus believed hard to solve exactly, a great deal of research has been devoted to estimating the failure probability. In this paper, we give a fully polynomial randomized approximation scheme that, given any n-vertex graph with specified failure probabilities, computes in time polynomial in n and $1/\epsilon$ an estimate for the failure probability that is accurate to within a relative error of $1\pm\epsilon$ with high probability. We also give a deterministic polynomial approximation scheme for the case of small failure probabilities. Some extensions to evaluating probabilities of $k$-connectivity, strong connectivity in directed Eulerian graphs and $r$-way disconnection, and to evaluating the Tutte polynomial are also described. This version of the paper corrects several errata that appeared in the previous journal publication [D. R. Karger, SIAM J. Comput., 29 (1999), pp. 492--514].

SIGEST

Theoretical computer science, itself a huge area, has contributed terminology and methodology to many other fields of applied mathematics and computing. Almost everyone in science has heard of the question "Does P = NP?" and of "NP-hard" problems. The SIAM Journal on Computing (SICOMP) represents a wide array of topics in theoretical computer science, the mathematical aspects of computer science. and nonnumerical computing. This issue's SIGEST selection, "A Randomized Fully Polynomial Time Approximation Scheme for the All-Terminal Network Reliability Problem," by David R. Karger, first appeared in SICOMP, volume 29 (1999), and has already been honored as one of the three SIAM Paper Prize winners in 2000. This paper is especially appropriate for SIGEST because it not only solves an important, formerly open problem (about which more later) but also presents a concise summary of $\sharp$P-complete problems, polynomial approximation schemes, and fully polynomial randomized approximation schemes.Counting problems, such as "Given a graph, how many distinct Hamiltonian circuits does it have?", involve a determination of the number of solutions, rather than requiring all solutions to be displayed. Such problems arise directly in statistical physics and indirectly in discrete approximations to certain continuous problems. For counting problems, the complexity class $\sharp$P is analogous to the class NP for decision problems. Given that a problem is $\sharp$P-hard, so that efficient algorithms guaranteed to find exact counts are unlikely, a fall-back position is to devise approximation algorithms, techniques guaranteed to find solutions that are provably near-exact, with a precise definition of "near." Development of approximation schemes can be extremely difficult, requiring both algorithm design and complexity-theoretic proofs of lower bounds.The problem addressed in Karger's paper is the classic all-terminal network reliability problem---determining the probability that a network will disconnect when links may fail independently. The paper presents a fully polynomial time approximation scheme that estimates the probability of failure, which is generally harder to approximate, as well as more interesting, than the probability of staying connected (except for very unreliable networks). Even readers unfamiliar with theoretical computer science will be able to appreciate the overall motivation for Karger's approach and the combination of results needed for the proofs. In addition, the paper provides a nice introduction to the strategies of approximation algorithms and to random sampling techniques applied to counting problems in graphs.

A Randomized Fully Polynomial Time Approximation Scheme for the All-Terminal Network Reliability Problem

The classic all-terminal network reliability problem posits a graph, each of whose edges fails independently with some given probability. The goal is to determine the probability that the network b...

A Randomized Fully Polynomial Time Approximation Scheme for the All-Terminal Network Reliability Problem

The classic all-terminal network reliability problem posits a graph, each of whose edges fails independently with some given probability. The goal is to determine the probability that the network becomes disconnected due to edge failures. This problem has obvious applications in the design of communication networks. Since the problem is $\SP$-complete and thus believed hard to solve exactly, a great deal of research has been devoted to estimating the failure probability. In this paper, we give a fully polynomial randomized approximation scheme that, given any n-vertex graph with specified failure probabilities, computes in time polynomial in n and $1/\epsilon$ an estimate for the failure probability that is accurate to within a relative error of $1\pm\epsilon$ with high probability. We also give a deterministic polynomial approximation scheme for the case of small failure probabilities. Some extensions to evaluating probabilities of k-connectivity, strong connectivity in directed Eulerian graphs and r-way disconnection, and to evaluating the Tutte polynomial are also described.

Finding the most vital edges with respect to the number of spanning trees

A most vital edge of a graph (w.r.t. the spanning trees) is an edge whose deletion most drastically decreases the number of spanning trees. We present an algorithm for determining the most vital edges based on Kirchoff's matrix-tree theorem whose asymptotic time-complexity can be reduced to that of the fastest matrix multiplication routine, currently O(n/sup 2.376/). The foundation for this approach is a more general algorithm for directed graphs for counting the rooted spanning arborescences containing each of the arcs of a digraph. A network can be modeled as a probabilistic graph. Under one such model proposed by Kel'mans, the all-terminal network reliability, maximizing the number of spanning trees is critical to maximizing reliability when edges are very unreliable. For this model, the most vital edges characterize the locations where an improvement of the reliability of the link most improves the reliability of the network.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Combining vertex decompositions with factoring in an all-terminal network reliability algorithm

The all-terminal network reliability problem is to determine the probability that all nodes can communicate in a network in which edges may fail independently of each other. One well known type of algorithm for treating this problem involves repeated deletion and contraction of edges in the network. These are commonly referred to as “factoring” algorithms and are especially effective on sparse graphs. On the other hand, a quite different approach known as the Buzacott algorithm is widely regarded as the most efficient algorithm for complete graphs. We demonstrate a manner in which higher degree vertex decomposition can be utilized within a factoring framework. This approach is compared to conventional factoring algorithms and the Buzacott algorithm from the standpoint of arithmetic operations performed and, in the case of the factoring algorithms, the search structures encountered during execution. Examples suggest that the higher degree vertex decompositions frequently reduce computational complexity to a greater extent than can be achieved by concentrating instead on an edge selection strategy to be used in factoring.

Chromatic polynomials and network reliability

In this paper, we introduce and study an extension of the chromatic polynomial of a graph. The new polynomial, determined by a graph G and a subset K of points of G, coincides with the classical chromatic polynomial when K is the set of all points of G. The main theorems in the present paper include analogues of the standard axiomatic characterization and Whitney's topological characterizations of the chromatic polynomial, and the theorem of Stanley relating the chromatic polynomial to the number of acylic of G. The work in this paper was stimulated by important connections between the chromatic polynomial and the all-terminal network reliability problem, and by recent work of Boesch, Satyanarayana, and Suffel on a graph invariant related to the K-terminal reliability problem. Several of the Boesch, Satyanarayana, and Suffel are derived as corollaries to the main theorems of the present paper.

Computational Complexity of Network Reliability Analysis: An Overview

This paper presents an overview of results related to the computational complexity of network reliability analysis problems. Network reliability analysis problems deal with the determination of reliability measures for stochastic networks. We show how these problems are related to the more familiar computational network problems of recognizing certain subnetworks, finding optimal subnetworks, and counting certain subnetworks. We use these relationships to show that the k-terminal, the 2-terminal, and the all-terminal network reliability analysis problems are at least as hard as the renowned set of computationally difficult problems, NP-Complete. Finally, we discuss the impact of these results on how one should approach problem solving in this area.