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This book presents a new approach to the analysis of networks, which emphasizes how one can compress a network while preserving all information relative to the network's spectrum. Besides these compression techniques, the authors introduce a number of other isospectral transformations and demonstrate how, together, these methods can be applied to gain new results in a number of areas. This includes the stability of time-delayed and non time-delayed dynamical networks, eigenvalue estimation, pseudospectra analysis and the estimation of survival probabilities in open dynamical systems. The theory of isospectral transformations, developed in this text, can be readily applied in any area that involves the analysis of multidimensional systems and is especially applicable to the analysis of network dynamics. This book will be of interest to Mathematicians, Physicists, Biologists, Engineers and to anyone who has an interest in the dynamics of networks.
List of contents
Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks.- Isospectral Matrix Reductions.- Dynamical Networks and Isospectral Graph Reductions.- Stability of Dynamical Networks.- Improved Eigenvalue Estimates.- Pseudospectra and Inverse Pseudospectra.- Improved Estimates of Survival Probabilities.
About the author
Leonid Bunimovich is a Professor of Mathematics and Director of the ABC Math Program at Georgia Institute of Technology. Professor Bunimovich's fields of interest include: Dynamical Systems, Ergodic Theory, Statistical Mechanics, Space-Time Chaos, Intermittency and Coherent Structures in Extended Systems, Geophysical Hydrodynamics, Mathematical Biology, Quantum Chaos, Waves in Nonhomogeneous Media, Lattice Gases, Cellular Automata, Percolation, Limit Theorems for Chaotic Dynamical Systems, Neuroscience, Operations Research.
Benjamin Webb is currently a visiting Assistant Professor in the Department of Mathematics at Brigham Young University. He has earned his PhD in Mathematics at the Georgia Institute of Technology.
Summary
This book presents a new approach to the analysis of networks, which emphasizes how one can compress a network while preserving all information relative to the network's spectrum. Besides these compression techniques, the authors introduce a number of other isospectral transformations and demonstrate how, together, these methods can be applied to gain new results in a number of areas. This includes the stability of time-delayed and non time-delayed dynamical networks, eigenvalue estimation, pseudospectra analysis and the estimation of survival probabilities in open dynamical systems. The theory of isospectral transformations, developed in this text, can be readily applied in any area that involves the analysis of multidimensional systems and is especially applicable to the analysis of network dynamics. This book will be of interest to Mathematicians, Physicists, Biologists, Engineers and to anyone who has an interest in the dynamics of networks.
Additional text
“An interesting fact is that a matrix with functional entries can have the same spectrum as a larger matrix with real entries. This is the foundation of the monograph under review. … The monograph is of significance for those specializing in the study of linear algebra applications to dynamical systems, and in particular for graduate students and researchers interested in the stability of dynamical networks.” (Juan Gonzalo Barajas-Ramírez, Mathematical Reviews, June, 2015)
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"An interesting fact is that a matrix with functional entries can have the same spectrum as a larger matrix with real entries. This is the foundation of the monograph under review. ... The monograph is of significance for those specializing in the study of linear algebra applications to dynamical systems, and in particular for graduate students and researchers interested in the stability of dynamical networks." (Juan Gonzalo Barajas-Ramírez, Mathematical Reviews, June, 2015)
