Albin N. Mathematics of Networks. Modulus Theory and Convex Optimization 2026

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Albin N. Mathematics of Networks. Modulus Theory and Convex Optimization 2026

Textbook in PDF format 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. The book is divided into several sections. Section I focuses on establishing the mathematical notation and theory for the various interconnecting subjects to be woven together, including the essentials of graph theory, electrical networks, spectral graph theory, and random walks. Section II provides a short but self-contained introduction to the theory of convex optimization. It is by no means intended to be comprehensive but contains sufficient detail to provide context and background for understanding the theory of modulus. Section III provides a comprehensive treatment of the fundamental theory of modulus on graphs, offering a basic understanding of what it means to talk about the modulus of a family of objects on a graph. Sections IV and V then discuss special features of the theory as applied to specific families. Section IV applies the theory to families of walks, which generalizes important concepts in graph metrics and electrical networks. Section V applies the theory to the family of spanning trees, yielding connections to several fascinating combinatorial concepts. Finally, Section VI provides a brief overview of the algorithms and computational techniques that can be used to experiment with modulus. Examples in Python. 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