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A weighted communicability measure applied to complex brain networks

JOURNAL OF THE ROYAL SOCIETY INTERFACE 2009年第33期6卷 411-414页

作者： Crofts, Jonathan J. Higham, Desmond J. Univ Strathclyde Glasgow G1 1XH Lanark Scotland

Recent advances in experimental neuroscience allow non-invasive studies of the white matter tracts in the human central nervous system, thus making available cutting-edge brain anatomical data describing these global connectivity patterns. Through magnetic resonance imaging, this non-invasive 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 real-valued connectivity data, thereby avoiding the need to discretize the corresponding adjacency matrix, i.e. 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.

关键词： matrix functions network science neuroscience unsupervised classification

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ON THE CONSTRUCTION AND PROPERTIES OF m-STEP METHODS FOR FDEs

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2015年第2期37卷 A653-A675页

作者： Aceto, Lidia Magherini, Cecilia Novati, Paolo Univ Pisa Dipartimento Matemat I-56127 Pisa Italy Univ Padua Dipartimento Matemat I-35121 Padua Italy

In this paper we consider the numerical solution of fractional differential equations by means of m-step recursions. The construction of such formulas can be obtained in many ways. Here we study a technique based on the rational approximation of the generating functions of fractional backward differentiation formulas (FBDFs). Accurate approximations lead to the definition of methods which simulate the underlying FBDF, with important computational advantages. Numerical experiments are presented.

关键词： fractional derivatives matrix functions contour integral approximation Gauss-Jacobi rule

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Computing functions of very large matrices with small TT/QTT ranks by quadrature formulas

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2020年第0期370卷 112663-000页

作者： Bertaccini, D. Durastante, F. Univ Roma Tor Vergata Dipartimento Matemat Rome Italy CNR Ist Applicaz Calcolo Mauro Picone Rome Italy CNR Ist Applicaz Calcolo Mauro Picone Naples Italy

The computation of matrix functions using quadrature formulas and rational approximations of very large structured matrices using tensor trains (TT), and quantized tensor trains (QTT) is considered here. The focus is on matrices with a small TT/QTT rank. Some analysis of the error produced by the use of the TT/QTT representation and the underlying approximation formula used is also provided. Promising experiments on exponential, power, Mittag-Leffler and logarithm function of multilevel Toeplitz matrices, that are among those which generate a low TT/QTT rank representation, are also provided, confirming that the proposed approach is feasible. (C) 2019 Elsevier B.V. All rights reserved.

关键词： matrix functions Quadrature formulas Tensor trains TT-format AMEn algorithm

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Communicability cosine distance: similarity and symmetry in graphs/networks

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COMPUTATIONAL & APPLIED MATHEMATICS 2024年第1期43卷 1-34页

作者： Estrada, Ernesto CSIC Inst Cross Disciplinary Phys & Complex Syst IFISC UIB Palma De Mallorca Spain

A distance based on the exponential kernel of the adjacency matrix of a graph and representing how well two vertices connect to each other in a graph is defined and studied. This communicability cosine distance (CCD) is a Euclidean spherical distance accounting for the cosine of the angles spanned by the position vectors of the graph vertices in this space. The Euclidean distance matrix (EDM) of CCD is used to quantify the similarity between vertices in graphs and networks as well as to define a local vertex invariant-a closeness centrality measure, which discriminate very well vertices in small graphs. It allows to distinguish all nonidentical vertices, also characterizing all identity (asymmetric) graphs-those having only the identity automorphism-among all connected graphs of up to 9 vertices. It also characterizes several other classes of identity graphs. We also study real-world networks in term of both the discriminating power of the new centrality on their vertices as well as in ranking their vertices. We analyze some dictionary networks as well as the network of copurshasing of political books, remarking some of the main advantages of the new approaches studied here.

关键词： Communicability matrix functions Graph automorphism Centrality measures Graph symmetry Vertex similarity

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PREDICTING TRIADIC CLOSURE IN NETWORKS USING COMMUNICABILITY DISTANCE functions

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SIAM JOURNAL ON APPLIED MATHEMATICS 2015年第4期75卷 1725-1744页

作者： Estrada, Ernesto Arrigo, Francesca Univ Strathclyde Dept Math & Stat Glasgow G1 1XQ Lanark Scotland Univ Insubria Dept Sci & High Technol I-22100 Como Italy

We propose a communication-driven mechanism for predicting triadic closure in complex networks. It is mathematically formulated on the basis of communicability distance functions that account for the quality of communication between nodes in the network. We study 25 real-world networks and show that the proposed method correctly predicts 20% of triadic closures in these networks, in contrast to the 7.6% predicted by a random mechanism. We also show that the communication-driven method outperforms the random mechanism in explaining the clustering coefficient, average path length, and average communicability. The new method also displays some interesting features with regards to optimizing communication in networks.

关键词： network analysis triangles triadic closure communicability distance adjacency matrix matrix functions

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Optimizing network robustness via Krylov subspaces

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ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS 2024年第1期58卷 131-155页

作者： Massei, Stefano Tudisco, Francesco Univ Pisa Dept Math Pisa Italy Univ Edinburgh Sch Math Edinburgh Scotland Univ Edinburgh Maxwell Inst Edinburgh Scotland Gran Sasso Sci Inst Laquila Italy

We consider the problem of attaining either the maximal increase or reduction of the robustness of a complex network by means of a bounded modification of a subset of the edge weights. We propose two novel strategies combining Krylov subspace approximations with a greedy scheme and an interior point method employing either the Hessian or its approximation computed via the limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm (L-BFGS). The paper discusses the computational and modeling aspects of our methodology and illustrates the various optimization problems on networks that can be addressed within the proposed framework. Finally, in the numerical experiments we compare the performances of our algorithms with state-of-the-art techniques on synthetic and real-world networks.

关键词： Network optimization low-rank approximation graph robustness optimization matrix functions Krylov-based optimization

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Computing Constrained Cramer-Rao Bounds

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2012年第10期60卷 5543-5548页

作者： Tune, Paul Univ Adelaide Sch Math Sci Adelaide SA 5005 Australia

We revisit the problem of computing submatrices of the Cramer-Rao bound (CRB), which lower bounds the variance of any unbiased estimator of a vector parameter theta. We explore iterative methods that avoid direct inversion of the Fisher information matrix, which can be computationally expensive when the dimension of theta is large. The computation of the bound is related to the quadratic matrix program, where there are highly efficient methods for solving it. We present several methods, and show that algorithms in prior work are special instances of existing optimization algorithms. Some of these methods converge to the bound monotonically, but in particular, algorithms converging nonmonotonically are much faster. We then extend the work to encompass the computation of the CRB when the Fisher information matrix is singular and when the parameter theta is subject to constraints. As an application, we consider the design of a data streaming algorithm for network measurement.

关键词： Cramer-Rao bound Fisher information matrix functions optimization quadratic matrix program

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A technique for improving the computation of functions of triangular matrices

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2022年第12期99卷 2449-2465页

作者： Cardoso, Joao R. Sadeghi, Amir Coimbra Polytech ISEC Coimbra Portugal Univ Coimbra Dept Math Ctr Math Coimbra Portugal Islamic Azad Univ Parand & Robat Karim Branch Dept Math Tehran Iran

We propose a simple technique that, if combined with algorithms for computing functions of triangular matrices, can make them more efficient. Basically, such a technique consists in a specific scaling similarity transformation that reduces the departure from normality of a triangular matrix, thus decreasing its norm and in general its function condition number. It can easily be extended to non-triangular matrices, provided that it is combined with algorithms involving a prior Schur decomposition. Situations where the technique should be used or not will be discussed in detail. Special attention is devoted to particular algorithms like the inverse scaling and squaring to the matrix logarithm or inverse cosine and the scaling and squaring to the matrix exponential. The advantages of our proposal are supported by theoretical results and illustrated with numerical experiments, involving matrices of small, medium and large size.

关键词： Schur decomposition matrix functions matrix logarithm matrix square roots matrix inverse cosine fractional powers of a matrix

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Exponentially Convergent Trapezoidal Rules to Approximate Fractional Powers of Operators

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JOURNAL OF SCIENTIFIC COMPUTING 2022年第2期91卷 1-18页

作者： Aceto, Lidia Novati, Paolo Univ Piemonte Orientale Dipartimento Sci & Innovaz Tecnol Viale T Michel 11 I-15121 Alessandria Italy Univ Trieste Dipartimento Matemat & Geosci Via Valerio 12-1 I-34127 Trieste Italy

In this paper we are interested in the approximation of fractional powers of self-adjoint positive operators. Starting from the integral representation of the operators, we apply the trapezoidal rule combined with a double-exponential transform of the integrand function. In this work we show how to improve the existing error estimates for the scalar case and also extend the analysis to operators. We report some numerical experiments to show the reliability of the estimates obtained.

关键词： matrix functions Double-exponential transform Trapezoidal rule Fractional Laplacian

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A novel two-stage strategy for robust H_∞ filter design of uncertain discrete-time systems

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JOURNAL OF THE FRANKLIN INSTITUTE-ENGINEERING AND APPLIED MATHEMATICS 2016年第13期353卷 3040-3056页

作者： Han, Kezhen Feng, Jian Northeastern Univ Sch Informat Sci & Engn Shenyang 110819 Peoples R China

This paper studies the robust Ho filter design problem for discrete-time linear system with polytopic uncertainty. Differing from existing ideas in the literature to reduce filtering conservatism, the present article proposes a less conservative two-stage algorithm to handle this issue by stepwisely recovering and utilizing matrix variable information of analysis condition. This algorithm is formulated by virtue of the combined matrix inequality with a prefixed binary choice signal, which is composed of direct design condition with nonlinear matrix variable inequality and its corresponding linearized one: In general;a more relaxed matrix variable structure is derived based on improved scalar parameter approach from the perspective of eigenvalue theory, which is then applied to obtain less conservative filter design condition with linearized matrix variable inequality in the first stage of provided algorithm. With some solved variables in stage one, that nonlinear matrix variable inequality based design condition is transformed into linear version, and in the second stage it will be used to further optimize the filtering performance level via picking up the lost matrix variable information of stage one. Finally, the advantages of the given algorithm are clearly illustrated by two examples. (C) 2016 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

关键词： DISCRETE-time systems ROBUST statistics UNCERTAINTY VARIABLES (Mathematics) matrix functions

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