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Full Description
Mathematics of Networks: Modulus Theory and Convex Optimization explores the question: "What can be learned by adapting the theory of p-modulus (and related continuum analysis concepts) to discrete graphs?" This book navigates the rich landscape of p-modulus on graphs, demonstrating how this theory elegantly connects concepts from graph theory, probability, and convex optimization.
This book is ideal for anyone seeking a deeper understanding of the theoretical foundations of network analysis and applied graph theory. It serves as an excellent primary text or reference for graduate and advanced undergraduate courses across multiple disciplines, including mathematics, data science, and engineering, particularly those focusing on network analysis, applied graph theory, optimization, and related areas.
Features:
Accessible to students with a solid foundation in multivariable calculus and linear algebra.
Broad interdisciplinary appeal, relevant to mathematics, data science, and engineering curricula.
Numerous engaging exercises.
Contents
Section I The Mathematics of Networks 1 Introduction to Graph Theory 2 Electrical Networks 3 The Laplacian 4 The Language of Probability 5 Basic theory of Markov chains 6 Connections 7 Mixing times Section II Optimization Problems on Graphs 8 Introduction to optimization 9 Continuous optimization Section III The Basic Theory of Modulus 10 Modulus on graphs 11 Dependence on the parameters 12 Duality for modulus 13 Probabilistic interpretation and blocking duality Section IV Families of Paths 14 Connecting Families 15 Modulus on Planar Maps 16 Modulus metrics Section V The Family of Spanning Trees 17 Loops and Trees 18 Spanning Tree Modulus Section VI Algorithms for Modulus 19 Algorithms for modulus Bibliography Index
