This volume considers resistance large graphs which are connected, undirected, and weighted. Such networks provide a discrete model for physical processes in inhomogeneous media, including heat flow through perforated or porous media. These graphs also arise in data science, e.g., considering geometrizations of datasets, statistical inference, or the propagation of memes through social networks. Indeed, network analysis plays a crucial role in many other areas of data science and engineering. In these models, the weights on the edges may be understood as conductances, or as a measure of similarity. Resistance networks also arise in probability, as they correspond to a broad class of Markov chains.The present volume takes the nonstandard approach of analyzing resistance networks from the point of view of Hilbert space theory, where the inner product is defined in terms of Dirichlet energy. The resulting viewpoint emphasizes orthogonality over convexity and provides new insights into the connections between harmonic functions, operators, and boundary theory. Novel applications to mathematical physics are given, especially in regard to the question of self-adjointness of unbounded operators.New topics are covered in a host of areas accessible to multiple audiences, at both beginning and more advanced levels. This is accomplished by directly linking diverse applied questions to such key areas of mathematics as functional analysis, operator theory, harmonic analysis, optimization, approximation theory, and probability theory. Upper-level undergraduate and graduate students in mathematics, electrical engineering, probability statistics, theoretical computer science, data science, physics, and econometrics, who would like to get a deeper understanding of large network models. It includes students as well specialists from a host of neighboring areas that are different from analysis of large networks but related. Suitable for courses and self-study.

Operator Theory and Analysis of Infinite Networks: Theory and Applications (Contemporary Mathematics and Its Applications: Monographs, Expositions and Lecture Notes Book 7)

Operator Theory and Analysis of Infinite Networks: Theory and Applications (Contemporary Mathematics and Its Applications: Monographs, Expositions and Lecture Notes Book 7) PDF

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