Forest matrices around the Laplacian matrix

We study the matrices Q_k of in-forests of a weighted digraph G and their connections with the Laplacian matrix L of G. The (i,j) entry of Q_k is the total weight of spanning converging forests (in-forests) with k arcs such that i belongs to a tree rooted at j. The forest matrices, Q_k, can be calculated recursively and expressed by polynomials in the Laplacian matrix; they provide representations for the generalized inverses, the powers, and some eigenvectors of L. The normalized in-forest matrices are row stochastic; the normalized matrix of maximum in-forests is the eigenprojection of the Laplacian matrix, which provides an immediate proof of the Markov chain tree theorem. A source of these results is the fact that matrices Q_k are the matrix coefficients in the polynomial expansion of adj(a*I+L). Thereby they are precisely Faddeev's matrices for -L. Keywords: Weighted digraph; Laplacian matrix; Spanning forest; Matrix-forest theorem; Leverrier-Faddeev method; Markov chain tree theorem; Eigenprojection; Generalized inverse; Singular M-matrixComment: 19 pages, presented at the Edinburgh (2001) Conference on Algebraic Graph Theor

Effective partitioning method for computing weighted Moore-Penrose inverse

We introduce a method and an algorithm for computing the weighted Moore-Penrose inverse of multiple-variable polynomial matrix and the related algorithm which is appropriated for sparse polynomial matrices. These methods and algorithms are generalizations of algorithms developed in [M.B. Tasic, P.S. Stanimirovic, M.D. Petkovic, Symbolic computation of weighted Moore-Penrose inverse using partitioning method, Appl. Math. Comput. 189 (2007) 615-640] to multiple-variable rational and polynomial matrices and improvements of these algorithms on sparse matrices. Also, these methods are generalizations of the partitioning method for computing the Moore-Penrose inverse of rational and polynomial matrices introduced in [P.S. Stanimirovic, M.B. Tasic, Partitioning method for rational and polynomial matrices, Appl. Math. Comput. 155 (2004) 137-163; M.D. Petkovic, P.S. Stanimirovic, Symbolic computation of the Moore-Penrose inverse using partitioning method, Internat. J. Comput. Math. 82 (2005) 355-367] to the case of weighted Moore-Penrose inverse. Algorithms are implemented in the symbolic computational package MATHEMATICA

Spectral Simplicity of Apparent Complexity, Part I: The Nondiagonalizable Metadynamics of Prediction

Virtually all questions that one can ask about the behavioral and structural complexity of a stochastic process reduce to a linear algebraic framing of a time evolution governed by an appropriate hidden-Markov process generator. Each type of question---correlation, predictability, predictive cost, observer synchronization, and the like---induces a distinct generator class. Answers are then functions of the class-appropriate transition dynamic. Unfortunately, these dynamics are generically nonnormal, nondiagonalizable, singular, and so on. Tractably analyzing these dynamics relies on adapting the recently introduced meromorphic functional calculus, which specifies the spectral decomposition of functions of nondiagonalizable linear operators, even when the function poles and zeros coincide with the operator's spectrum. Along the way, we establish special properties of the projection operators that demonstrate how they capture the organization of subprocesses within a complex system. Circumventing the spurious infinities of alternative calculi, this leads in the sequel, Part II, to the first closed-form expressions for complexity measures, couched either in terms of the Drazin inverse (negative-one power of a singular operator) or the eigenvalues and projection operators of the appropriate transition dynamic.Comment: 24 pages, 3 figures, 4 tables; current version always at http://csc.ucdavis.edu/~cmg/compmech/pubs/sdscpt1.ht

ADDITIVE PROPERTIES OF THE DRAZIN INVERSE FOR MATRICES AND BLOCK REPRESENTATIONS: A SURVEY

In this paper, a review of a development of the Drazin inverse for the sum of two matrices has been given. Since this topic is closely related to the problem of finding the Drazin inverse of a 2x2 block matrix, the paper also offers a survey of this subject

On the Poisson equation for nonreversible Markov jump processes

We study the solution $V$ of the Poisson equation $LV + f=0$ where $L$ is the backward generator of an irreducible (finite) Markov jump process and $f$ is a given centered state function. Bounds on $V$ are obtained using a graphical representation derived from the Matrix Forest Theorem and using a relation with mean first-passage times. Applications include estimating time-accumulated differences during relaxation toward a steady nonequilibrium regime

Essays on the economics of networks

Networks (collections of nodes or vertices and graphs capturing their linkages) are a common object of study across a range of fields includ- ing economics, statistics and computer science. Network analysis is often based around capturing the overall structure of the network by some reduced set of parameters. Canonically, this has focused on the notion of centrality. There are many measures of centrality, mostly based around statistical analysis of the linkages between nodes on the network. However, another common approach has been through the use of eigenfunction analysis of the centrality matrix. My the- sis focuses on eigencentrality as a property, paying particular focus to equilibrium behaviour when the network structure is fixed. This occurs when nodes are either passive, such as for web-searches or queueing models or when they represent active optimizing agents in network games. The major contribution of my thesis is in the applica- tion of relatively recent innovations in matrix derivatives to centrality measurements and equilibria within games that are function of those measurements. I present a series of new results on the stability of eigencentrality measures and provide some examples of applications to a number of real world examples

Recurrent neural networks for solving matrix algebra problems

The aim of this dissertation is the application of recurrent neural networks (RNNs) to solving some problems from a matrix algebra with particular reference to the computations of the generalized inverses as well as solving the matrix equations of constant (timeinvariant) matrices. We examine the ability to exploit the correlation between the dynamic state equations of recurrent neural networks for computing generalized inverses and integral representations of these generalized inverses. Recurrent neural networks are composed of independent parts (sub-networks). These sub-networks can work simultaneously, so parallel and distributed processing can be accomplished. In this way, the computational advantages over the existing sequential algorithms can be attained in real-time applications. We investigate and exploit an analogy between the scaled hyperpower family (SHPI family) of iterative methods for computing the matrix inverse and the discretization of Zhang Neural Network (ZNN) models. A class of ZNN models corresponding to the family of hyperpower iterative methods for computing the generalized inverses on the basis of the discovered analogy is defined. The Matlab Simulink implementation of the introduced ZNN models is described in the case of scaled hyperpower methods of the order 2 and 3. We present the Matlab Simulink model of a hybrid recursive neural implicit dynamics and give a simulation and comparison to the existing Zhang dynamics for real-time matrix inversion. Simulation results confirm a superior convergence of the hybrid model compared to Zhang model

ITERATIONS FOR APPROXIMATING LIMIT REPRESENTATIONS OF GENERALIZED INVERSES

Our underlying motivation is the iterative method for the implementation of the limit representation of the Moore-Penrose inverse$\lim\limits _{\alpha \to 0} \left( \alpha I+A^*A\right) ^{-1} A^*$ from [\v Zukovski, Lipcer, On recurent computation of normal solutions of linear algebraic equations, \v Z. Vicisl. Mat. i Mat. Fiz. 12 (1972), 843--857] and[\v Zukovski, Lipcer, On computation pseudoinverse matrices, \v Z. Vicisl. Mat. i Mat. Fiz. 15 (1975), 489--492].  The iterative process for the implementation of the general limit formula$\lim\limits _{\alpha \to 0}(\alpha I+R^*S) ^{-1}R^*$was defined in [P.S. Stanimirovi\'c, Limit representations of generalized inverses and related methods, Appl. Math. Comput. 103 (1999), 51--68].In this paper we develop an improvement of this iterative process.The iterative method defined in such a way is able to produce the result in a predefined number of iterative steps. Convergence properties of defined iterations are further investigated

On discrete-time dissipative port-Hamiltonian (descriptor) systems

Port-Hamiltonian (pH) systems have been studied extensively for linear continuous-time dynamical systems. This manuscript presents a discrete-time pH descriptor formulation for linear, completely causal, scattering passive dynamical systems based on the system coefficients. The relation of this formulation to positive and bounded real systems and the characterization via positive semidefinite solutions of Kalman-Yakubovich-Popov inequalities is also studied.Comment: 30 pages, 3 figure