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66
The scaling and squaring method for the matrix exponential revisited
 SIAM REV
, 2009
"... The calculation of the matrix exponential e A maybeoneofthebestknownmatrix problems in numerical computation. It achieved folk status in our community from the paper by Moler and Van Loan, “Nineteen Dubious Ways to Compute the Exponential of a Matrix, ” published in this journal in 1978 (and revisit ..."
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Cited by 96 (20 self)
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The calculation of the matrix exponential e A maybeoneofthebestknownmatrix problems in numerical computation. It achieved folk status in our community from the paper by Moler and Van Loan, “Nineteen Dubious Ways to Compute the Exponential of a Matrix, ” published in this journal in 1978 (and revisited in this journal in 2003). The matrix exponential is utilized in a wide variety of numerical methods for solving differential equations and many other areas. It is somewhat amazing given the long history and extensive study of the matrix exponential problem that one can improve upon the best existing methods in terms of both accuracy and efficiency, but that is what the SIGEST selection in this issue does. “The Scaling and Squaring Method for the Matrix Exponential Revisited ” by N. Higham, originally published in the SIAM Journal on Matrix Analysis and Applications in 2005, applies a new backward error analysis to the commonly used scaling and squaring method, as well as a new rounding error analysis of the Padé approximant of the scaled matrix. The analysis shows, and the accompanying experimental results verify, that a Padé approximant of a higher order than currently used actually results in a more accurate
Multilayer networks
 TOOL FOR MULTILAYER ANALYSIS AND VISUALIZATION OF NETWORKS 17 OF 18
, 2014
"... In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time, and include other types of complications. Such systems include multiple subsystems and layers of connectivity, and it is impo ..."
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Cited by 30 (7 self)
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In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time, and include other types of complications. Such systems include multiple subsystems and layers of connectivity, and it is important to take such “multilayer” features into account to try to improve our understanding of complex systems. Consequently, it is necessary to generalize “traditional ” network theory by developing (and validating) a framework and associated tools to study multilayer systems in a comprehensive fashion. The origins of such efforts date back several decades and arose in multiple disciplines, and now the study of multilayer networks has become one of the most important directions in network science. In this paper, we discuss the history of multilayer networks (and related concepts) and review the exploding body of work on such networks. To unify the disparate terminology in the large body of recent work, we discuss a general framework for multilayer networks, construct a dictionary
Network Properties Revealed Through Matrix Functions
, 2008
"... The newly emerging field of Network Science deals with the tasks of modelling, comparing and summarizing large data sets that describe complex interactions. Because pairwise affinity data can be stored in a twodimensional array, graph theory and applied linear algebra provide extremely useful tools. ..."
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Cited by 30 (3 self)
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The newly emerging field of Network Science deals with the tasks of modelling, comparing and summarizing large data sets that describe complex interactions. Because pairwise affinity data can be stored in a twodimensional array, graph theory and applied linear algebra provide extremely useful tools. Here, we focus on the general concepts of centrality, communicability and betweenness, each of which quantifies important features in a network. Some recent work in the mathematical physics literature has shown that the exponential of a network’s adjacency matrix can be used as the basis for defining and computing specific versions of these measures. We introduce here a general class of measures based on matrix functions, and show that a particular case involving a matrix resolvent arises naturally from graphtheoretic arguments. We also point out connections between these measures and the quantities typically computed when spectral methods are used for data mining tasks such as clustering and ordering. We finish with computational examples showing the new matrix resolvent version applied to real networks.
Communicability across evolving networks
 PHYSICAL REVIEW E
, 2010
"... Many natural and technological applications generate time ordered sequences of networks, defined over a fixed set of nodes; for example timestamped information about ‘who phoned who ’ or ‘who came into contact with who ’ arise naturally in studies of communication and the spread of disease. Concept ..."
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Cited by 21 (10 self)
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Many natural and technological applications generate time ordered sequences of networks, defined over a fixed set of nodes; for example timestamped information about ‘who phoned who ’ or ‘who came into contact with who ’ arise naturally in studies of communication and the spread of disease. Concepts and algorithms for static networks do not immediately carry through to this dynamic setting. For example, suppose A and B interact in the morning, and then B and C interact in the afternoon. Information, or disease, may then pass from A to C, but not vice versa. This subtlety is lost if we simply summarize using the daily aggregate network given by the chain ABC. However, using a natural definition of a walk on an evolving network, we show that classic centrality measures from the static setting can be extended in a computationally convenient manner. In particular, communicability indices can be computed to summarize the ability of each node to broadcast and receive information. The computations involve basic operations in linear algebra, and the asymmetry caused by time’s arrow is captured naturally through the noncommutativity of matrixmatrix multiplication. Illustrative examples are given for both synthetic and realworld communication data sets. We also discuss the use of the new centrality measures for realtime monitoring and prediction.
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 ..."
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Cited by 18 (3 self)
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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.
Community landscapes: an integrative approach to determine overlapping network module hierarchy, identify key nodes and predict network dynamics
 PLoS One
, 2010
"... Background: Network communities help the functional organization and evolution of complex networks. However, the development of a method, which is both fast and accurate, provides modular overlaps and partitions of a heterogeneous network, has proven to be rather difficult. Methodology/Principal Fin ..."
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Cited by 17 (1 self)
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Background: Network communities help the functional organization and evolution of complex networks. However, the development of a method, which is both fast and accurate, provides modular overlaps and partitions of a heterogeneous network, has proven to be rather difficult. Methodology/Principal Findings: Here we introduce the novel concept of ModuLand, an integrative method family determining overlapping network modules as hills of an influence functionbased, centralitytype community landscape, and including several widely used modularization methods as special cases. As various adaptations of the method family, we developed several algorithms, which provide an efficient analysis of weighted and directed networks, and (1) determine pervasively overlapping modules with high resolution; (2) uncover a detailed hierarchical network structure allowing an efficient, zoomin analysis of large networks; (3) allow the determination of key network nodes and (4) help to predict network dynamics. Conclusions/Significance: The concept opens a wide range of possibilities to develop new approaches and applications
A WEIGHTED COMMUNICABILITY MEASURE APPLIED TO COMPLEX BRAIN NETWORKS
, 2008
"... Abstract. Recent advances in experimental neuroscience allow noninvasive studies of the white matter tracts in the human central nervous system, thus making available cuttingedge brain anatomical data describing these global connectivity patterns. Via magnetic resonance imaging, this noninvasive ..."
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Cited by 14 (4 self)
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Abstract. Recent advances in experimental neuroscience allow noninvasive studies of the white matter tracts in the human central nervous system, thus making available cuttingedge brain anatomical data describing these global connectivity patterns. Via magnetic resonance imaging, this noninvasive technique is able to infer a snapshot of the cortical network within the living human brain. Here, we report on the initial success of a new weighted network communicability measure in distinguishing local and global differences between diseased patients and controls. This approach builds on recent advances in network science, where an underlying connectivity structure is used as a means to measure the ease with which information can flow between nodes. One advantage of our method is that it deals directly with the realvalued connectivity data, thereby avoiding the need to discretise the corresponding adjacency matrix, that is, to round weights up to 1 or down to 0, depending upon some threshold value. Experimental results indicate that the new approach is able to extract biologically relevant features that are not immediately apparent from the raw connectivity data. 1.