Applied and Computational Mathematics Division

The reliability polynomial gives the probability that a graph remains connected given that each edge in it can fail independently with an identical probability. While in general determining the coefficients of this polynomial is #P-complete, we give a randomized algorithm for approximating its coefficients. When compared to the known approximation method of Colbourn, Debroni and Myrvold, our method empirically shows a much faster rate of convergence. * Official contribution of the National Institute of Standards and Technology; not subject to copyright in the United States.

We investigate the relative computational power of three classes of pointer algorithms by comparing time and space usage. These classes include two from Tarjan's original paper and another class which we introduce which is related to another long standing problem.

The ability to create random models of real networks is useful for understanding the interactions in these systems. Several researchers have proposed modeling complex networks by using the node degree distribution, the most popular being a power-law distribution. Recent work by Li et al. introduced the S metric as a metric to characterize the structure of networks with power-law distributions. In this paper, we examine some of the practical difficulties of producing random graphs with a given degree sequence and an approximate S value. We give a solution for this problem that we have had success using in our research.

We define a class of matroids A for which a fully polynomial randomized approximation scheme (fpras) exists for counting the number of bases of the matroids. We then show that as the number of elements in a matroid increases, the probability that a matroid belongs to A goes to 1. We thus provide a fpras for counting the number of bases that applies to almost all matroids. * Official contribution of the National Institute of Standards and Technology; not subject to copyright in the United States.

A number of important combinatorial counting problems can be reformulated into the problem of counting the number of leaf nodes on a tree. Since the basic leaf-counting problem is #P-complete, there is strong evidence that no polynomial time algorithm exists for this general problem. Thus, we propose a randomized approximation scheme for this problem, and then empirically compare its convergence rate with the classic method of Knuth. We then give an application of our scheme by introducing a new algorithm for estimating the number of bases of a matroid with an independence oracle. * Official contribution of the National Institute of Standards and Technology; not subject to copyright in the United States.

One of the most important parts of designing an expert system is elicitation of the expert's knowledge. This knowledge usually consists of facts and rules. Eliciting these rules and facts is relatively easy, the more complicated task is assigning weights (numerical or interval-valued degrees of belief) to di erent statements from the knowledge base. Expert

Simulation of many dynamic real world systems such as the Internet and social networks requires developing dynamic models for the underlying networks in these systems. Currently, there is a large body of work devoted towards determining the underlying mechanisms that create these networks, but the resulting models have not realistically captured many of the important structural characteristics when compared with real world examples. Towards creating more realistic dynamic models, we propose a method of structurally constructing models of an evolving network. We then conduct a series of computational experiments in modeling the evolution of the autonomous system (AS) topology of the Internet to test the effectiveness of our approach.

Recent research has shown that while many complex networks follow a power-law distribution for their node degrees, it is not sufficient to model these networks based only on their degree distribution. In order to better distinguish between these networks, the metric s was introduced to measure how interconnected the hub nodes are in a network.

The reliability polynomial gives the probability that a graph remains connected given that each edge in it can fail independently with an identical probability. While in general determining the coefficients of this polynomial is #P-complete, we give a randomized algorithm for approximating its coefficients. When compared to the known approximation method of C. J. Colbourn, B. M. Debroni and W. J. Myrvold [Estimating the coefficients of the reliability polynomial. Numerical mathematics and computing, 17th Manitoba Conf., Winnipeg Can. 1987, Congr. Numerantium 62, 217–223 (1988; Zbl 0662.90036)], our method empirically shows a much faster rate of convergence.

The ability to create random models of real networks is useful for understanding the interactions in these systems. Several researchers have proposed modeling complex networks by using the node degree distribution, the most popular being a power-law distribution. Recent work by Li et al. introduced the S metric as a metric to characterize the structure of networks with power-law distributions. In this paper, we examine some of the practical difficulties of producing random graphs with a given degree sequence and an approximate S value. We give a solution for this problem that we have had success using in our research.

We define a class of matroids A for which a fully polynomial randomized approximation scheme (fpras) exists for counting the number of bases of the matroids. We then show that as the number of elements in a matroid increases, the probability that a matroid belongs to A goes to 1. We thus provide a fpras for counting the number of bases that applies to almost all matroids. * Official contribution of the National Institute of Standards and Technology; not subject to copyright in the United States.

Relationships between entities in many complex systems, such as the Internet and social networks, have a natural hierarchical organization. Understanding these inherent hierarchies is essential for creating models of these systems. Thus, there is a recent body of research concerning the extraction of hierarchies from networks. We propose a new method for modeling hierarchies through extracting the affiliations of the network. From these affiliations, we construct a lattice of the relationships between nodes. A principal advantage of our approach is that any overlapping community structures of the nodes within the network have a natural representation within the lattice. We then show an example of our method using a real data set.

We examine the problem of creating random realizations of very large degree sequences. Although fast in practice, the Markov chain Monte Carlo (MCMC) method for selecting a realization has limited usefulness for creating large graphs because of memory constraints. Instead, we focus on sequential importance sampling (SIS) schemes for random graph creation. A difficulty with SIS schemes is assuring that they terminate in a reasonable amount of time. We introduce a new sampling method by which we guarantee termination while achieving speed comparable to the MCMC method.

We define a class of matroids A for which a fully polynomial randomized approximation scheme (fpras) exists for counting the number of bases of the matroids. We then show that as the number of elements in a matroid increases, the probability that a matroid belongs to A goes to 1. We thus provide a fpras for counting the number of bases that applies to almost all matroids. * Official contribution of the National Institute of Standards and Technology; not subject to copyright in the United States.

We investigate the relative computational power of three classes of pointer algorithms by comparing time and space usage. These classes include two from Tarjan's original paper and another class which we introduce which is related to another long standing problem.

A number of important combinatorial counting problems can be reformulated into the problem of counting the number of leaf nodes on a tree. Since the basic leaf-counting problem is #P-complete, there is strong evidence that no polynomial time algorithm exists for this general problem. Thus, we propose a randomized approximation scheme for this problem, and then empirically compare its convergence rate with the classic method of Knuth. We then give an application of our scheme by introducing a new algorithm for estimating the number of bases of a matroid with an independence oracle. * Official contribution of the National Institute of Standards and Technology; not subject to copyright in the United States.

The reliability polynomial gives the probability that a graph remains connected given that each edge in it can fail independently with an identical probability. While in general determining the coefficients of this polynomial is #P-complete, we give a randomized algorithm for approximating its coefficients. When compared to the known approximation method of Colbourn, Debroni and Myrvold, our method empirically shows a much faster rate of convergence. * Official contribution of the National Institute of Standards and Technology; not subject to copyright in the United States.

We examine the problem of creating random realizations of very large degree sequences. Although fast in practice, the Markov chain Monte Carlo (MCMC) method for selecting a realization has limited usefulness for creating large graphs because of memory constraints. Instead, we focus on sequential importance sampling (SIS) schemes for random graph creation. A difficulty with SIS schemes is assuring that they terminate in a reasonable amount of time. We introduce a new sampling method by which we guarantee termination while achieving speed comparable to the MCMC method.