Results 1  10
of
29
DECAY PROPERTIES OF SPECTRAL PROJECTORS WITH APPLICATIONS TO ELECTRONIC STRUCTURE
, 2010
"... Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our theory leads to a rigorous proof of the ..."
Abstract

Cited by 18 (3 self)
 Add to MetaCart
(Show Context)
Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our theory leads to a rigorous proof of the exponential offdiagonal decay (‘nearsightedness’) for the density matrix of gapped systems at zero electronic temperature in both orthogonal and nonorthogonal representations, thus providing a firm theoretical basis for the possibility of linear scaling methods in electronic structure calculations for nonmetallic systems. Our theory also allows us to treat the case of density matrices for arbitrary systems at finite electronic temperature, including metals. Other possible applications are also discussed.
Network centrality in the human functional connectome
 Cereb. Cortex
, 2012
"... The network architecture of functional connectivity within the human brain connectome is poorly understood at the voxel level. Here, using resting state functional magnetic resonance imaging data from 1003 healthy adults, we investigate a broad array of network centrality measures to provide novel i ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
The network architecture of functional connectivity within the human brain connectome is poorly understood at the voxel level. Here, using resting state functional magnetic resonance imaging data from 1003 healthy adults, we investigate a broad array of network centrality measures to provide novel insights into connectivity within the wholebrain functional network (i.e., the functional connectome). We first assemble and visualize the voxelwise (4 mm) functional connectome as a functional network. We then demonstrate that each centrality measure captures different aspects of connectivity, highlighting the importance of considering both global and local connectivity properties of the functional connectome. Beyond ‘‘detecting functional hubs,’ ’ we treat centrality as measures of functional connectivity within the brain connectome and demonstrate their reliability and phenotypic correlates (i.e., age and sex). Specifically, our analyses reveal agerelated decreases in degree
TOTAL COMMUNICABILITY AS A CENTRALITY MEASURE
"... Abstract. We examine node centrality measures based on the notion of total communicability, defined in terms of the row sums of matrix functions of the adjacency matrix of the network. Our main focus is on the matrix exponential and the resolvent, which have natural interpretations in terms of walks ..."
Abstract

Cited by 9 (5 self)
 Add to MetaCart
Abstract. We examine node centrality measures based on the notion of total communicability, defined in terms of the row sums of matrix functions of the adjacency matrix of the network. Our main focus is on the matrix exponential and the resolvent, which have natural interpretations in terms of walks on the underlying graph. While such measures have been used before for ranking nodes in a network, we show that they can be computed very rapidly even in the case of large networks. Furthermore, we propose the (normalized) total sum of node communicabilities as a useful measure of network connectivity. Extensive numerical studies are conducted in order to compare this centrality measure with the closely related ones of subgraph centrality [E. Estrada and J. A. RodríguezVelázquez, Phys. Rev. E, 71 (2005), 056103] and Katz centrality [L. Katz, Psychometrica, 18 (1953), pp. 39–43]. Both synthetic and realworld networks are used in the computations.
MultiScale Link Prediction
"... The automated analysis of social networks has become an important problem due to the proliferation of social networks, such as LiveJournal, Flickr and Facebook. The scale of these social networks is massive and continues to grow rapidly. An important problem in social network analysis is proximity e ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
(Show Context)
The automated analysis of social networks has become an important problem due to the proliferation of social networks, such as LiveJournal, Flickr and Facebook. The scale of these social networks is massive and continues to grow rapidly. An important problem in social network analysis is proximity estimation that infers the closeness of different users. Link prediction, in turn, is an important application of proximity estimation. However, many methods for computing proximity measures have high computational complexity and are thus prohibitive for largescale link prediction problems. One way to address this problem is to estimate proximity measures via lowrank approximation. However, a single lowrank approximation may not be sufficient to represent the behavior of the entire network. In this paper, we propose MultiScale Link Prediction (MSLP), a framework for link prediction, which can handle massive networks. The basic idea of MSLP is to construct lowrank approximations of the network at multiple scales in an efficient manner. To achieve this, we propose a fast treestructured approximation algorithm. Based on this approach, MSLP combines predictions at multiple scales to make robust and accurate predictions. Experimental results on reallife datasets with more than a million nodes show the superior performance and scalability of our method.
Network Analysis via Partial Spectral Factorization and Gauss Quadrature
, 2012
"... Abstract. Largescale networks arise in many applications. It is often of interest to be able to identify the most important nodes of a network or to ascertain the ease of traveling between nodes. These and related quantities can be determined by evaluating expressions of the form uT f(A)w, where A ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
(Show Context)
Abstract. Largescale networks arise in many applications. It is often of interest to be able to identify the most important nodes of a network or to ascertain the ease of traveling between nodes. These and related quantities can be determined by evaluating expressions of the form uT f(A)w, where A is the adjacency matrix that represents the graph of the network, f is a nonlinear function, such as the exponential function, and u and w are vectors, for instance, axis vectors. This paper describes a novel technique for determining upper and lower bounds for expressions uT f(A)w when A is symmetric and bounds for many vectors u and w are desired. The bounds are computed by first evaluating a lowrank approximation of A, which is used to determine rough bounds for the desired quantities for all nodes. These rough bounds indicate for which vectors u and w more accurate bounds should be computed with the aid of Gausstype quadrature rules. This hybrid approach is cheaper than only using Gausstype rules to determine accurate upper and lower bounds in the common situation when it is not known a priori for which vectors u and w accurate bounds for uT f(A)w should be computed. Several computed examples, including an application to software engineering, illustrate the performance of the hybrid method.
A Catalogue of Software for Matrix Functions. Version 1.0
, 2014
"... A catalogue of software for computing matrix functions and their Fréchet derivatives is presented. For a wide variety of languages and for software ranging from commercial products to open source packages we describe what matrix function codes are available and which algorithms they implement. ..."
Abstract

Cited by 4 (4 self)
 Add to MetaCart
A catalogue of software for computing matrix functions and their Fréchet derivatives is presented. For a wide variety of languages and for software ranging from commercial products to open source packages we describe what matrix function codes are available and which algorithms they implement.
Higher order Fréchet derivatives of matrix functions and the level2 condition number
 Manchester Institute for Mathematical Sciences, The University of Manchester
, 2013
"... Abstract. The Fréchet derivative Lf of a matrix function f: C n×n → Cn×n controls the sensitivity of the function to small perturbations in the matrix. While much is known about the properties of Lf and how to compute it, little attention has been given to higher order Fréchet derivatives. We der ..."
Abstract

Cited by 4 (4 self)
 Add to MetaCart
Abstract. The Fréchet derivative Lf of a matrix function f: C n×n → Cn×n controls the sensitivity of the function to small perturbations in the matrix. While much is known about the properties of Lf and how to compute it, little attention has been given to higher order Fréchet derivatives. We derive sufficient conditions for the kth Fréchet derivative to exist and be continuous in its arguments and we develop algorithms for computing the kth derivative and its Kronecker form. We analyze the level2 absolute condition number of a matrix function (“the condition number of the condition number”) and bound it in terms of the second Fréchet derivative. For normal matrices and the exponential we show that in the 2norm the level1 and level2 absolute condition numbers are equal and that the relative condition numbers are within a small constant factor of each other. We also obtain an exact relationship between the level1 and level2 absolute condition numbers for the matrix inverse and arbitrary nonsingular matrices, as well as a weaker connection for Hermitian matrices for a class of functions that includes the logarithm and square root. Finally, the relation between the level1 and level2 condition numbers is investigated more generally through numerical
Block Gauss and antiGauss quadrature with application to networks
 SIAM J. Matrix Anal. Appl
"... Abstract. Approximations of matrixvalued 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 blockvector W ∈ Rm×k. ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
(Show Context)
Abstract. Approximations of matrixvalued 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 blockvector 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 matrixvalued measure. This paper generalizes antiGauss quadrature rules, introduced by Laurie for realvalued measures, to matrixvalued measures, and shows that under suitable conditions pairs of block Gauss and block antiGauss rules provide upper and lower bounds for the entries of the desired matrixvalued function. Extensions to matrixvalued functions of the form WT f(A)V, where A ∈ Rm×m may be nonsymmetric, and the matrices V,W ∈ Rm×k satisfy V TW = Ik are also discussed. Approximations of the latter functions are computed by applying a few steps of the nonsymmetric block Lanczos method to A with initial blockvectors 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 antiGauss 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.
ESTIMATING THE CONDITION NUMBER OF THE FRÉCHET DERIVATIVE OF A MATRIX FUNCTION∗
, 2013
"... Abstract. The Fréchet derivative Lf of a matrix function f: Cn×n → Cn×n is used in a variety of applications and several algorithms are available for computing it. We define a condition number for the Fréchet derivative and derive upper and lower bounds for it that differ by at most a factor 2. Fo ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
(Show Context)
Abstract. The Fréchet derivative Lf of a matrix function f: Cn×n → Cn×n is used in a variety of applications and several algorithms are available for computing it. We define a condition number for the Fréchet derivative and derive upper and lower bounds for it that differ by at most a factor 2. For a wide class of functions we derive an algorithm for estimating the 1norm condition number that requires O(n3) flops given O(n3) flops algorithms for evaluating f and Lf; in practice it produces estimates correct to within a factor 6n. Numerical experiments show the new algorithm to be much more reliable than a previous heuristic estimate of conditioning.