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

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 non-classical, like probability theory, measure theory, tensor algebras, and semidefinite optimization).