Reliability Polynomial Research Articles

Wang–Landau sampling for estimation of the reliability of physical networks

Modern physical networks, for example in communication and transportation, can be interpreted as directed graphs. Network models are used to identify the probability that given nodes are connected, and therefore the effect of a failure at a given link. This is essential for network design, optimization, and reliability. In this study, we investigated three alternative ensembles for estimating network reliability using the Wang–Landau algorithm. The first performed random walks on a structure function having two possible states: connected and disconnected. The second used random walks on a reliability polynomial. The third combined random walks with the average of connecting probabilities. The accuracy and limitations of the three ensembles were compared by estimating the reliability of three network models: a bridge network, a ladder-type network, and a dodecahedron network. The simulation results showed that the use of a random walk on a structure function failed to produce estimates when applied to highly reliable networks in any of the three network types. The other two approaches performed efficiently for bridge or ladder-type networks at any level of network reliability. The random walk on a probability space using the 1∕t algorithm was the only ensemble that was able to yield accurate estimates for a dodecahedron network, though even this failed at the highest level of network reliability. The other two methods failed to converge within 108 Monte Carlo trials. The use of the average of connecting probabilities required a shorter computation time when applied to a large network. Methods that can reduce variance for large, highly reliable networks require further investigation.

Four Input Sorter Good, Larger Ones Not So Good

Our own fresh results have suggested that the smallest size optimal sorting networks/graphs might be the most appropriate entities (building blocks) for designing highly reliable computing systems. Sorting networks correspond to particular sorting algorithms, while their associated connectivity graphs can embody minimal two-terminal networks. Relying on such concepts, one can associate a reliability polynomial to any sorting network. Here, we are going to expand on results we have recently reported on sorting networks by examining a few slightly larger optimal sorting networks. Comparing the two-terminal reliability polynomials associated to these optimal sorting connectivity graphs with those of Moore-Shannon hammocks of the same size will follow. In-depth comparisons will be done using both classical as well as novel figures-of-merit targeting the reliability enhancements of networks when performing computations. The results reported here will shed light on our previous findings, confirming that the optimal sorting network of four inputs is ideal for designing highly reliable computing systems, but also that its advantages, although of theoretical importance, are only marginal and fading quickly. In particular, analyzing optimal sorting networks of larger number of inputs reveals that hammocks are catching up with optimal sorting networks at five and six inputs, while clearly overtaking the optimal sorting network of seven inputs. All these simulations and detailed comparisons provide compelling arguments for why hammock networks are the ones we should rely upon (at more than one level) for 2D reliable computations (including for the current quantum computing quest).

MARKOV CHAIN METHOD FOR COMPUTING THE RELIABILITY OF HAMMOCK NETWORKS

In this paper, we develop a new method for evaluating the reliability polynomial of a hammock network. The method is based on a homogeneous absorbing Markov chain and provides the exact reliability for networks of width less than 5 and arbitrary length. Moreover, it produces a lower bound for the reliability polynomial for networks of width greater than or equal to 5. To investigate how sharp this lower bound is, we compare our method with other approximation methods and it proves to be the most accurate in terms of absolute as well as relative error. Using the fundamental matrix, we also calculate the average time to absorption, which provides the mean length of a network that is expected to work.

Roots of two‐terminal reliability polynomials

AbstractAssume that the vertices of a graph G are always operational, but the edges of G are operational independently with probability p ∈ [0, 1]. For fixed vertices s and t, the two‐terminal reliability of G is the probability that the operational subgraph contains an (s, t)‐path, while the all‐terminal reliability of G is the probability that the operational subgraph contains a spanning tree. Both reliabilities are polynomials in p, and have very similar behavior in many respects. However, unlike all‐terminal reliability polynomials, little is known about the roots of two‐terminal reliability polynomials. In a variety of ways, we shall show that the nature and location of the roots of two‐terminal reliability polynomials have significantly different properties than those held by roots of the all‐terminal reliability polynomials.

Most reliable two‐terminal graphs with node failures

AbstractA two‐terminal graph is an undirected graph with two specified target vertices. If each nontarget vertex of a two‐terminal graph fails independently with the same fixed probability (and edges and target vertices are perfectly reliable), the two‐terminal node reliability is the probability that the target vertices are in the same connected component in the induced subgraph of all operational nodes. A two‐terminal graph is uniformly most reliable if its node reliability polynomial is greater than or equal to that of all other two‐terminal graphs with the same fixed number of vertices, n, and edges, m. In this paper, we show that there is always a uniformly most reliable two‐terminal graph. Furthermore, with the additional restriction that the distance between the target vertices is at least three, we completely classify which values of n and m produce a uniformly most reliable graph and which do not.

A Sequential Importance Sampling Algorithm for Counting Linear Extensions

In recent decades, a number of profound theorems concerning approximation of hard counting problems have appeared. These include estimation of the permanent, estimating the volume of a convex polyhedron, and counting (approximately) the number of linear extensions of a partially ordered set. All of these results have been achieved using probabilistic sampling methods, specifically Monte Carlo Markov Chain (MCMC) techniques. In each case, a rapidly mixing Markov chain is defined that is guaranteed to produce, with high probability, an accurate result after only a polynomial number of operations. Although of polynomial complexity, none of these results lead to a practical computational technique, nor do they claim to. The polynomials are of high degree and a non-trivial amount of computing is required to get even a single sample. Our aim in this article is to present practical Monte Carlo methods for one of these problems, counting linear extensions. Like related work on estimating the coefficients of the reliability polynomial, our technique is based on improving the so-called Knuth counting algorithm by incorporating an importance function into the node selection technique giving a sequential importance sampling (SIS) method. We define and report performance on two importance functions.

Rational roots of all‐terminal reliability

AbstractGiven a connected graph G whose vertices are perfectly reliable and whose edges each fail independently with probability q ∈ [0, 1], the (all‐terminal) reliability of G is the probability that the resulting subgraph of operational edges contains a spanning tree (this probability is always a polynomial in q). The location of the roots of reliability polynomials has been well studied, with particular interest in finding those with the largest moduli. In this paper, we will discuss a related problem—among all reliability polynomials of graphs on n vertices, what can we say about the rational roots? We prove that (for n ≥ 2), the rational roots are −1, − 1/2, − 1/3,…, − 1/(n − 1), 1. Moreover, we show that for n ≥ 3, the root of minimum modulus among all graphs of order n is rational, and determine all roots of smallest moduli and the corresponding graphs. Finally, we provide the first nontrivial mathematical property that distinguishes, via reliability, the class of simple graphs (i.e., those without loops and multiple edges) from that of graphs in general.

Full Hermite interpolation of the reliability of a hammock network

Although the hammock networks were introduced more than sixty years ago, there is no general formula of the associated reliability polynomial. Using the full Hermite interpolation polynomial, we propose an approximation for the reliability polynomial of a hammock network of arbitrary size. In the second part of the paper, we provide combinatorial formulas for the first two non-zero coefficients of the reliability polynomial.

Classes of uniformly most reliable graphs for all-terminal reliability

The all-terminal reliability polynomial of a graph G is the probability that G remains connected if each edge is independently included with fixed probability p. Among all graphs with a fixed number of n vertices and m edges, a graph is uniformly most reliable if its reliability polynomial is greater than or equal to all other graphs on the same number of vertices and edges for every value of p∈[0,1]. We find a new class of graphs that are uniformly most reliable. Specifically, we show that when n≥5 is odd and m is either n2−n+12 or n2−n+32, there exists a unique uniformly most reliable graph, and we give its construction.

On the reliability roots of simplicial complexes and matroids

Assume that the vertices of a graph G are always operational, but the edges of G fail independently with probability q∈[0,1]. The all-terminal reliability of G is the probability that the resulting subgraph is connected. The all-terminal reliability can be formulated into a polynomial in q, and it was conjectured that all the roots of (nonzero) reliability polynomials fall inside the closed unit disk. It has since been shown that there exist some connected graphs which have their reliability roots outside the closed unit disk, but these examples seem to be few and far between, and the roots are only barely outside the disk. In this paper we generalize the notion of reliability to simplicial complexes and matroids and investigate when the roots fall inside the closed unit disk. We show that such is the case for matroids of rank 3 and paving matroids of rank 4. We also prove that the reliability roots of shellable complexes are dense in the complex plane, and that the real reliability roots of any matroid lie in [−1,0)∪{1}. Finally, we also show that the reliability roots of thickenings of the Fano matroid can lie outside the unit disk.

A Recursive Formula for the Reliability of a r-Uniform Complete Hypergraph and Its Applications

The reliability polynomial R(S,p) of a finite graph or hypergraph S=(V,E) gives the probability that the operational edges or hyperedges of S induce a connected spanning subgraph or subhypergraph, respectively, assuming that all (hyper)edges of S fail independently with an identical probability q=1-p. In this paper, we investigate the probability that the hyperedges of a hypergraph with randomly failing hyperedges induce a connected spanning subhypergraph. The computation of the reliability for (hyper)graphs is an NP-hard problem. We provide recurrence relations for the reliability of r-uniform complete hypergraphs with hyperedge failure. Consequently, we determine and calculate the number of connected spanning subhypergraphs with given size in the r-uniform complete hypergraphs.

How Reliable are Compositions of Series and Parallel Networks Compared with Hammocks?

A classical problem in computer/network reliability is that of identifying simple, regular and repetitive building blocks (motifs) which yield reliability enhancements at the system-level. Over time, this apparently simple problem has been addressed by various increasingly complex methods. The earliest and simplest solutions are series and parallel structures. These were followed by majority voting and related schemes. For the most recent solutions, which are also the most involved (e.g., those based on Harary and circulant graphs), optimal reliability has been proven under particular conditions. 
 Here, we propose an alternate approach for designing reliable systems as repetitive compositions of the simplest possible structures. More precisely, our two motifs (basic building blocks) are: two devices in series, and two devices in parallel. Therefore, for a given number of devices (which is a power of two) we build all the possible compositions of series and parallel networks of two devices. For all of the resulting twoterminal networks, we compute exactly the reliability polynomials, and then compare them with those of size-equivalent hammock networks. The results show that compositions of the two simplest motifs are not able to surpass size-equivalent hammock networks in terms of reliability. Still, the algorithm for computing the reliability polynomials of such compositions is linear (extremely effcient), as opposed to the one for the size-equivalent hammock networks, which is exponential. Interestingly, a few of the compositions come extremely close to size-equivalent hammock networks with respect to reliability, while having fewer wires.a

Sixty Years of Network Reliability

The study of network reliability started in 1956 with a groundbreaking paper by E.F. Moore and C.E. Shannon. They introduced a probabilistic model of network reliability, where the nodes of the network were considered to be perfectly reliable, and the links or edges could fail independently with a certain probability. The problem is to determine the probability that the network remains connected under these conditions. If all the edges have the same probability of failing, this leads to the so-called reliability polynomial of the network. Sixty years later, a lot of research has accumulated on this topic, and many variants of the original problem have been investigated. We review the basic concepts and results, as well as some recent developments in this area, and we outline some important research directions.

Graphs models and algorithms for reliability assessment of coherent and non-coherent systems

In this article, we propose new models and algorithms for the reliability assessment of systems relying on concepts of graphs theory. These developments exploit the order relation on the set of system components’ states which is graphically represented by the Hasse diagram. The monotony property of the reliability structure function of coherent systems allows us to obtain an ordered graph from the Hasse diagram. This ordered graph represents all the system states and it can be obtained from only the knowledge of the system tie-sets. First of all, this model gives a new way for the research of a minimal disjoint Boolean polynomial, and, second, it is able to directly find the system reliability without resorting to an intermediate Boolean polynomial. Browsing the paths from the minimal tie-sets to the maxima of the ordered graph and using a weight associated with each node, we are able to propose a new algorithm to directly obtain the reliability polynomial by the research of sub-graphs representing eligible monomials. This approach is then extended to non-coherent systems thanks to the introduction of a new concept of terminal tie-sets. These algorithms are applied to some case studies, for both coherent and non-coherent real systems, and the results compared with those computed using standard reliability block diagram or fault tree models validate the proposed approach. Formal definitions of used graphs and of developed algorithms are also given, making their software implementation easy and efficient.

On uniformly most reliable two‐terminal graphs

A two‐terminal graph is an undirected graph G with vertex set V(G), edge set E(G), and two specified target vertices in V(G). If each edge of such a graph operates independently with the same fixed probability p, the two‐terminal reliability is the probability that there exists a path between the target vertices. A two‐terminal graph is uniformly most reliable if its reliability polynomial is greater than or equal to that of all other two‐terminal graphs with the same fixed number of vertices, n, and edges, m. In this article, we present specific values of n and m for which no uniformly most reliable two‐terminal simple graph exists, as well as values of n and m for which there does exist a uniformly most reliable two‐terminal simple graph.

On the Exact Reliability Enhancements of Small Hammock Networks

In this paper, we study small hammock networks, more precisely the 29 hammock networks presented by Moore and Shannon in their prescient paper from 1956, where this type of network was introduced. We will first of all review the concept of the reliability polynomial, and define hammock networks, and emphasize some of their properties. We will then review the state of the art from three different angles: theoretical, algorithmic, and design oriented. These will show that results obtained over many years advocate very strongly for the small hammock networks. Because array-based designs (e.g., FinFETs) seem to be a perfect fit for small hammocks (as are nanoelectromechanical systems or NEMS), we have decided to start investigating them closely. The analyses we pursue here are exact, the coefficients of the reliability polynomials being determined using our own program. The reliability polynomials for non-trivial hammock networks are reported here as functions of the probability ( $p$ ) that the device is closed. As far as we know, this is the first time they have appeared in this form. Our purpose in determining exact results is a practical one, as we foresee that small hammock networks could play a key role in any array-based design. That is why their reliability enhancements are evaluated thoroughly, using both classical and non-standard cost functions. Finally, conclusions are followed by a longer list of future directions of research, some of them practical while others more theoretical.

Problems on multivariate reliability polynomial

The original results include: (i) homogenization of a reliability polynomial; (ii) compact hypersurfaces attached to homogeneous polynomials; (iii) an affine diffeomorphism that preserves a reliability polynomial; (iv) duality of networks via a diffeomorphism; (v) straight line segments in constant level sets associated to a reliability polynomial.

Fourientations and the Tutte polynomial

A fourientation of a graph is a choice for each edge of the graph whether to orient that edge in either direction, leave it unoriented, or biorient it. Fixing a total order on the edges and a reference orientation of the graph, we investigate properties of cuts and cycles in fourientations which give trivariate generating functions that are generalized Tutte polynomial evaluations of the form (k+m)n-1(k+l)gTαk+βl+mk+m,γk+l+δmk+l\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} (k+m)^{n-1}(k+l)^gT\\left( \\frac{\\alpha k + \\beta l + m}{k+m},\\; \\frac{\\gamma k + l + \\delta m}{k+l}\\right) \\end{aligned}$$\\end{document}for alpha ,gamma in {0,1,2} and beta , delta in {0,1}. We introduce an intersection lattice of 64 cut–cycle fourientation classes enumerated by generalized Tutte polynomial evaluations of this form. We prove these enumerations using a single deletion–contraction argument and classify axiomatically the set of fourientation classes to which our deletion–contraction argument applies. This work unifies and extends earlier results for fourientations due to Gessel and Sagan (Electron J Combin 3(2):Research Paper 9, 1996), results for partial orientations due to Backman (Adv Appl Math, forthcoming, 2014. arXiv:1408.3962), and Hopkins and Perkinson (Trans Am Math Soc 368(1):709–725, 2016), as well as results for total orientations due to Stanley (Discrete Math 5:171–178, 1973; Higher combinatorics (Proceedings of NATO Advanced Study Institute, Berlin, 1976). NATO Advanced Study Institute series, series C: mathematical and physical sciences, vol 31, Reidel, Dordrecht, pp 51–62, 1977), Las Vergnas (Progress in graph theory (Proceedings, Waterloo silver jubilee conference 1982), Academic Press, New York, pp 367–380, 1984), Greene and Zaslavsky (Trans Am Math Soc 280(1):97–126, 1983), and Gioan (Eur J Combin 28(4):1351–1366, 2007), which were previously unified by Gioan (2007), Bernardi (Electron J Combin 15(1):Research Paper 109, 2008), and Las Vergnas (Tutte polynomial of a morphism of matroids 6. A multi-faceted counting formula for hyperplane regions and acyclic orientations, 2012. arXiv:1205.5424). We conclude by describing how these classes of fourientations relate to geometric, combinatorial, and algebraic objects including bigraphical arrangements, cycle–cocycle reversal systems, graphic Lawrence ideals, Riemann–Roch theory for graphs, zonotopal algebra, and the reliability polynomial.

Partial graph orientations and the Tutte polynomial

Gessel and Sagan investigated the Tutte polynomial, TG(x,y) using depth-first search, and applied their techniques to show that the number of acyclic partial orientations of a graph is 2m−n+1TG(3,1/2). We provide a short deletion-contraction proof of this result and demonstrate that dually, the number of strongly connected partial orientations is 2n−1TG(1/2,3). We then prove that the number of partial orientations modulo cycle reversals is 2gTG(3,1) and the number of partial orientations modulo cut reversals is 2n−1TG(1,3). To prove these results, we introduce cut and cycle-minimal partial orientations which provide distinguished representatives for partial orientations modulo cut and cycle reversals, extending known representatives for full orientations introduced by Greene and Zaslavksy. We then introduce distinguished partial orientations representing a given indegree sequence. We utilize these partial orientations to derive the Ehrhart polynomial of the win vector polytope, and give a combinatorial interpretation of its volume, thus answering a question of Bartels, Mount, and Welsh. We conclude with edge chromatic generalizations of the quantities presented, which allow for a new interpretation of the reliability polynomial for all probabilities p with 0<p<1/2.

A recursive formula for the two‐edge connected and biconnected reliability of a complete graph

The reliability polynomial of a finite undirected graph gives the probability that the operational edges of induce a connected graph assuming that all edges of fail independently with identical probability . In this article we investigate the probability that the operational edges of a graph with randomly failing edges induce a biconnected or two‐edge connected subgraph, which corresponds to demands for redundancy or higher throughput in communication networks. The computation of the biconnected or two‐edge connected reliability for general graphs is computationally intractable (#P‐hard). We provide recurrence relations for biconnected and two‐edge connected reliability of complete graphs. As a consequence, we can determine the number of biconnected and two‐edge connected graphs with given order and size. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 69(4), 408–414 2017