931 research outputs found
SoftNet: A Package for the Analysis of Complex Networks
Identifying the most important nodes according to specific centrality indices is an important issue in network analysis. Node metrics based on the computation of functions of the adjacency matrix of a network were defined by Estrada and his collaborators in various papers. This paper describes a MATLAB toolbox for computing such centrality indices using efficient numerical algorithms based on the connection between the Lanczos method and Gauss-type quadrature rules
The geometric mean of two matrices from a computational viewpoint
The geometric mean of two matrices is considered and analyzed from a
computational viewpoint. Some useful theoretical properties are derived and an
analysis of the conditioning is performed. Several numerical algorithms based
on different properties and representation of the geometric mean are discussed
and analyzed and it is shown that most of them can be classified in terms of
the rational approximations of the inverse square root functions. A review of
the relevant applications is given
Spectral element methods: Algorithms and architectures
Spectral element methods are high-order weighted residual techniques for partial differential equations that combine the geometric flexibility of finite element methods with the rapid convergence of spectral techniques. Spectral element methods are described for the simulation of incompressible fluid flows, with special emphasis on implementation of spectral element techniques on medium-grained parallel processors. Two parallel architectures are considered: the first, a commercially available message-passing hypercube system; the second, a developmental reconfigurable architecture based on Geometry-Defining Processors. High parallel efficiency is obtained in hypercube spectral element computations, indicating that load balancing and communication issues can be successfully addressed by a high-order technique/medium-grained processor algorithm-architecture coupling
Block Gauss and anti-Gauss quadrature with application to networks
Approximations of matrix-valued functions of the form WT f(A)W, where A ∈Rm×m is symmetric, W ∈ Rm×k, with m large and k ≪ m, has orthonormal columns, and f is a function, can be computed by applying a few steps of the symmetric block Lanczos method to A with initial block-vector W ∈ Rm×k. Golub and Meurant have shown that the approximants obtained in this manner may be considered block Gauss quadrature rules associated with a matrix-valued measure. This paper generalizes anti-Gauss quadrature rules, introduced by Laurie for real-valued measures, to matrix-valued measures, and shows that under suitable conditions pairs of block Gauss and block anti-Gauss rules provide upper and lower bounds for the entries of the desired matrix-valued function. Extensions to matrix-valued functions of the form WT f(A)V , where A ∈ Rm×m may be nonsymmetric, and the matrices V, W ∈ Rm×k satisfy VT W = Ik also are discussed. Approximations of the latter functions are computed by applying a few steps of the nonsymmetric block Lanczos method to A with initial block-vectors V and W. We describe applications to the evaluation of functions of a symmetric or nonsymmetric adjacency matrix for a network. Numerical examples illustrate that a combination of block Gauss and anti-Gauss quadrature rules typically provides upper and lower bounds for such problems. We introduce some new quantities that describe properties of nodes in directed or undirected networks, and demonstrate how these and other quantities can be
computed inexpensively with the quadrature rules of the present paper
Updating and downdating techniques for optimizing network communicability
The total communicability of a network (or graph) is defined as the sum of
the entries in the exponential of the adjacency matrix of the network, possibly
normalized by the number of nodes. This quantity offers a good measure of how
easily information spreads across the network, and can be useful in the design
of networks having certain desirable properties. The total communicability can
be computed quickly even for large networks using techniques based on the
Lanczos algorithm.
In this work we introduce some heuristics that can be used to add, delete, or
rewire a limited number of edges in a given sparse network so that the modified
network has a large total communicability. To this end, we introduce new edge
centrality measures which can be used to guide in the selection of edges to be
added or removed.
Moreover, we show experimentally that the total communicability provides an
effective and easily computable measure of how "well-connected" a sparse
network is.Comment: 20 pages, 9 pages Supplementary Materia
Computation of generalized matrix functions
We develop numerical algorithms for the efficient evaluation of quantities
associated with generalized matrix functions [J. B. Hawkins and A. Ben-Israel,
Linear and Multilinear Algebra 1(2), 1973, pp. 163-171]. Our algorithms are
based on Gaussian quadrature and Golub--Kahan bidiagonalization. Block variants
are also investigated. Numerical experiments are performed to illustrate the
effectiveness and efficiency of our techniques in computing generalized matrix
functions arising in the analysis of networks.Comment: 25 paged, 2 figure
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