This volume considers resistance networks: 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.

Contents:

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  Preface

  
  


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  About the Authors

  
  


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  Acknowledgments

  
  


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  List of Figures

  
  


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  List of Symbols and Notation

  
  


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  Introduction

  
  


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  Resistance Networks

  
  


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  The Energy Hilbert Space

  
  


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  The Resistance Metric

  
  


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  Schoenberg–von Neumann Construction of the Energy Space Hε

  
  


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  The Laplacian on Hε

  
  


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  The ℓ² Theory of Δ and the Transfer Operator

  
  


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  The Boundary and Boundary Representation

  
  


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  Multiplication Operators on the Energy Space

  
  


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  Symmetric Pairs

  
  


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  The Dissipation Space HD

  
  


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  Probabilistic Interpretations

  
  


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  Spectral Comparisons

  
  


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  Examples and Applications

  
  


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  Lattice Networks

  
  


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  Infinite Trees

  
  


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  Application to Magnetism and Long-Range Order

  
  


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  Future Directions

  
  


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  Appendices:

  
  
  
  
  • Some Functional Analysis
  
  
  • Some Operator Theories
  
  
  • Navigation Aids for Operators and Spaces
  
  
  • A Guide to the Bibliography
  
  
  
  


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  Bibliography

  
  


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  Index

  
  

Readership: 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.

Key Features:

• For the first time, a detailed account of the theory of infinite networks, with numerous illustrations and explicit examples


• The text offers an insight-oriented approach offering immediacy and flexibility


• The topics are presented in a straightforward style, answering questions in the context of compelling examples


• Introducing also more advanced concepts. This approach motivates the more abstract theory via interesting applications


• This book lays the basic foundation for infinite networks and includes numerous applications, making it beneficial to mathematicians as well as to physicists and engineers


• The book includes guides for students and instructors, for classroom use, and for self-study