Mathematics of Networks: Modulus Theory and Convex Optimization is a book that seeks to answer the question: “What can be learned by adapting the theory of p-modulus (and related continuum analysis concepts) to discrete graphs?”
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.
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
Nathan Albin received a BA in Mathematics from the University of Hawaii and Hilo in 2001. Following a short stint as a software engineer, he attended graduate school at the University of Utah and received a PhD in Mathematics in 2006. He was then awarded an NSF Mathematical Sciences Postdoctoral Research Fellowship, hence spent one year researching at the University of Duisburg-Essen, before moving to Pasadena, CA to complete his postdoctoral research at Caltech. He was hired as an Assistant Professor of Mathematics in 2011 and was promoted to Associate Professor in 2016. His primary research interests include Applied Mathematics, Network Theory, Optimization Theory, and Numerical Analysis.
Pietro Poggi-Corradini grew up in Milan, Italy, and in 1992 received a Bachelor in Mathematics from the University of Milan. He then pursued his studies at the University of Washington in Seattle (USA) and graduated in 1996 with a PhD Thesis in Complex Dynamics and Operator Theory under Professor Don Marshall. Thereafter he became Whyburn Instructor at the University of Virginia and in 1998 joined the faculty at Kansas State University. He has been Full Professor at Kansas State University since 2005. His interests currently span Network Theory, Complex Analysis, Probability, and Applied Mathematics.
