Large Networks and Graph Limits

Front Cover
American Mathematical Soc., 2012 - Mathematics - 475 pages
Recently, it became apparent that a large number of the most interesting structures and phenomena of the world can be described by networks. To develop 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 non-classical, 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 theory of the algebraic theory of graph homomorphisms and of the analytic theory of graph limits. 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 This book is a comprehensive study of the active topic of graph limits and an updated account of its present status. It is a beautiful volume written by an outstanding mathematician who is also a great 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. 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 Laszlo Lovasz 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 Lovasz'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
 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

Very large networks
3
Large graphs in mathematics and physics
25
Notation and terminology
37
Graph homomorphisms
55
Graph algebras and homomorphism functions
83
Kernels and graphons
115
The cut distance
127
Szemerédi partitions
141
Extremal theory of dense graphs
281
Multigraphs and decorated graphs
317
Graphings
329
Convergence of bounded degree graphs
351
Right convergence of bounded degree graphs
367
On the structure of graphings
383
Algorithms for bounded degree graphs
397
Other combinatorial structures
415

Sampling
157
Convergence of dense graph sequences
173
Convergence from the right
201
On the structure of graphons
217
The space of graphons
239
Algorithms for large graphs and graphons
263
Appendix A Appendix
433
Bibliography
451
Author Index
465
Notation Index
473
Copyright

Common terms and phrases

Bibliographic information