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Hardcover ISBN:  9780821890851 
Product Code:  COLL/60 
List Price:  $99.00 
MAA Member Price:  $89.10 
AMS Member Price:  $79.20 
eBook ISBN:  9781470415839 
Product Code:  COLL/60.E 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
Hardcover ISBN:  9780821890851 
eBook ISBN:  9781470415839 
Product Code:  COLL/60.B 
List Price:  $188.00 $143.50 
MAA Member Price:  $169.20 $129.15 
AMS Member Price:  $150.40 $114.80 

Book DetailsColloquium PublicationsVolume: 60; 2012; 475 ppMSC: Primary 05; Secondary 90
Recently, it became apparent that a large number of the most interesting structures and phenomena of the world can be described by networks. Developing a mathematical theory of very large networks is an important challenge. This book describes one recent approach to this theory, the limit theory of graphs, which has emerged over the last decade. The theory has rich connections with other approaches to the study of large networks, such as “property testing” in computer science and regularity partition in graph theory. It has several applications in extremal graph theory, including the exact formulations and partial answers to very general questions, such as which problems in extremal graph theory are decidable. It also has less obvious connections with other parts of mathematics (classical and nonclassical, like probability theory, measure theory, tensor algebras, and semidefinite optimization).
This book explains many of these connections, first at an informal level to emphasize the need to apply more advanced mathematical methods, and then gives an exact development of the algebraic theory of graph homomorphisms and of the analytic theory of graph limits.
ReadershipGraduate students and research mathematicians interested in graph theory and its application to networks.

Table of Contents

Part 1. Large graphs: An informal introduction

Chapter 1. Very large networks

Chapter 2. Large graphs in mathematics and physics

Part 2. The algebra of graph homomorphisms

Chapter 3. Notation and terminology

Chapter 4. Graph parameters and connection matrices

Chapter 5. Graph homomorphisms

Chapter 6. Graph algebras and homomorphism functions

Part 3. Limits of dense graph sequences

Chapter 7. Kernels and graphons

Chapter 8. The cut distance

Chapter 9. Szemerédi partitions

Chapter 10. Sampling

Chapter 11. Convergence of dense graph sequences

Chapter 12. Convergence from the right

Chapter 13. On the structure of graphons

Chapter 14. The space of graphons

Chapter 15. Algorithms for large graphs and graphons

Chapter 16. Extremal theory of dense graphs

Chapter 17. Multigraphs and decorated graphs

Part 4. Limits of bounded degree graphs

Chapter 18. Graphings

Chapter 19. Convergence of bounded degree graphs

Chapter 20. Right convergence of bounded degree graphs

Chapter 21. On the structure of graphings

Chapter 22. Algorithms for bounded degree graphs

Part 5. Extensions: A brief survey

Chapter 23. Other combinatorial structures

Appendix A


Additional Material

Reviews

Written by an eminent expert as the first monograph on this topic, this book can be recommended to anybody working on large networks and their applications in mathematics, computer science, social sciences, biology, statistical physics or chip design.
Zentralblatt Math 
This is an amazing book: readable, deep, and lively. It sets out this emerging area, makes connections between old classical graph theory and graph limits, and charts the course of the future.
Persi Diaconis, Stanford University 
It is always exciting when a mathematical theory turns out to be connected to a variety of other topics. This is the case with the recently developed subject of graph limits, which exhibits tight relations with a wide range of areas including statistical physics, analysis, algebra, extremal graph theory, and theoretical computer science. The book Large Networks and Graph Limits contains a comprehensive study of this active topic and an updated account of its present status. The author, Laszls Lovasz, initiated the subject, and together with his collaborators has contributed immensely to its development during the last decade. This is a beautiful volume written by an outstanding mathematician who is also an excellent expositor.
Noga Alon, Tel Aviv University, Israel 
Modern combinatorics is by no means an isolated subject in mathematics, but has many rich and interesting connections to almost every area of mathematics and computer science. The research presented in Lovasz's book exemplifies this phenomenon by taking one of the most quintessentially combinatorial of objectsthe finite graphand through the process of taking limits of sequences of such graphs, reveals and clarifies connections to measure theory, analysis, statistical physics, metric geometry, spectral theory, property testing, algebraic geometry, and even Hilbert's tenth and seventeenth problems. Indeed, this book presents a wonderful opportunity for a student in combinatorics to explore other fields of mathematics, or conversely for experts in other areas of mathematics to become acquainted with some aspects of graph theory.
Terence Tao, University of California, Los Angeles, CA 
László Lovász has written an admirable treatise on the exciting new theory of graph limits and graph homomorphisms, an area of great importance in the study of large networks. It is an authoritative, masterful text that reflects Lovász's position as the main architect of this rapidly developing theory. The book is a must for combinatorialists, network theorists, and theoretical computer scientists alike.
Bela Bollobas, Cambridge University, UK


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Recently, it became apparent that a large number of the most interesting structures and phenomena of the world can be described by networks. Developing a mathematical theory of very large networks is an important challenge. This book describes one recent approach to this theory, the limit theory of graphs, which has emerged over the last decade. The theory has rich connections with other approaches to the study of large networks, such as “property testing” in computer science and regularity partition in graph theory. It has several applications in extremal graph theory, including the exact formulations and partial answers to very general questions, such as which problems in extremal graph theory are decidable. It also has less obvious connections with other parts of mathematics (classical and nonclassical, like probability theory, measure theory, tensor algebras, and semidefinite optimization).
This book explains many of these connections, first at an informal level to emphasize the need to apply more advanced mathematical methods, and then gives an exact development of the algebraic theory of graph homomorphisms and of the analytic theory of graph limits.
Graduate students and research mathematicians interested in graph theory and its application to networks.

Part 1. Large graphs: An informal introduction

Chapter 1. Very large networks

Chapter 2. Large graphs in mathematics and physics

Part 2. The algebra of graph homomorphisms

Chapter 3. Notation and terminology

Chapter 4. Graph parameters and connection matrices

Chapter 5. Graph homomorphisms

Chapter 6. Graph algebras and homomorphism functions

Part 3. Limits of dense graph sequences

Chapter 7. Kernels and graphons

Chapter 8. The cut distance

Chapter 9. Szemerédi partitions

Chapter 10. Sampling

Chapter 11. Convergence of dense graph sequences

Chapter 12. Convergence from the right

Chapter 13. On the structure of graphons

Chapter 14. The space of graphons

Chapter 15. Algorithms for large graphs and graphons

Chapter 16. Extremal theory of dense graphs

Chapter 17. Multigraphs and decorated graphs

Part 4. Limits of bounded degree graphs

Chapter 18. Graphings

Chapter 19. Convergence of bounded degree graphs

Chapter 20. Right convergence of bounded degree graphs

Chapter 21. On the structure of graphings

Chapter 22. Algorithms for bounded degree graphs

Part 5. Extensions: A brief survey

Chapter 23. Other combinatorial structures

Appendix A

Written by an eminent expert as the first monograph on this topic, this book can be recommended to anybody working on large networks and their applications in mathematics, computer science, social sciences, biology, statistical physics or chip design.
Zentralblatt Math 
This is an amazing book: readable, deep, and lively. It sets out this emerging area, makes connections between old classical graph theory and graph limits, and charts the course of the future.
Persi Diaconis, Stanford University 
It is always exciting when a mathematical theory turns out to be connected to a variety of other topics. This is the case with the recently developed subject of graph limits, which exhibits tight relations with a wide range of areas including statistical physics, analysis, algebra, extremal graph theory, and theoretical computer science. The book Large Networks and Graph Limits contains a comprehensive study of this active topic and an updated account of its present status. The author, Laszls Lovasz, initiated the subject, and together with his collaborators has contributed immensely to its development during the last decade. This is a beautiful volume written by an outstanding mathematician who is also an excellent expositor.
Noga Alon, Tel Aviv University, Israel 
Modern combinatorics is by no means an isolated subject in mathematics, but has many rich and interesting connections to almost every area of mathematics and computer science. The research presented in Lovasz's book exemplifies this phenomenon by taking one of the most quintessentially combinatorial of objectsthe finite graphand through the process of taking limits of sequences of such graphs, reveals and clarifies connections to measure theory, analysis, statistical physics, metric geometry, spectral theory, property testing, algebraic geometry, and even Hilbert's tenth and seventeenth problems. Indeed, this book presents a wonderful opportunity for a student in combinatorics to explore other fields of mathematics, or conversely for experts in other areas of mathematics to become acquainted with some aspects of graph theory.
Terence Tao, University of California, Los Angeles, CA 
László Lovász has written an admirable treatise on the exciting new theory of graph limits and graph homomorphisms, an area of great importance in the study of large networks. It is an authoritative, masterful text that reflects Lovász's position as the main architect of this rapidly developing theory. The book is a must for combinatorialists, network theorists, and theoretical computer scientists alike.
Bela Bollobas, Cambridge University, UK