Large Networks and Graph Limits by Laszlo Lovasz

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Large Networks and Graph Limits written by Laszlo Lovasz This is an other great mathematics book cover the following topics.

Part 1. Large graphs: an informal introduction

1. Very large networks
Huge networks everywhere, What to ask about them?, How to obtain information about them?, How to model them?, How to approximate them?, How to run algorithms on them?, Bounded degree graphs

2. Large graphs in mathematics and physics
Extremal graph theory, Statistical physics

Part 2. The algebra of graph homomorphisms

3. Notation and terminology
Basic notation, Graph theory, Operations on graphs

4. Graph parameters and connection matrices
Graph parameters and graph properties, Connection matrices, Finite connection rank

5. Graph homomorphisms
Existence of homomorphisms, Homomorphism numbers, What hom functions can express, Homomorphism and isomorphism, Independence of homomorphism functions, Characterizing homomorphism numbers, The structure of the homomorphism set

6. Graph algebras and homomorphism functions
Algebras of quantum graphs, Reflection positivity, Contractors and connectors, Algebras for homomorphism functions, Computing parameters with finite connection rank, The polynomial method

Part 3. Limits of dense graph sequences

7. Kernels and graphons
Kernels, graphons and stepfunctions, Generalizing homomorphisms, Weak isomorphism I, Sums and products, Kernel operators

8. The cut distance
The cut distance of graphs, Cut norm and cut distance of kernels, Weak and L1-topologies

9. Szemer´edi partitions
Regularity Lemma for graphs, Regularity Lemma for kernels, Compactness of the graphon space, Fractional and integral overlays, Uniqueness of regularity partitions

10. Sampling
W-random graphs, Sample concentration, Estimating the distance by sampling, The distance of a sample from the original, Counting Lemma, Inverse Counting Lemma, Weak isomorphism II

11. Convergence of dense graph sequences
Sampling, homomorphism densities and cut distance, Random graphs as limit objects, The limit graphon, Proving convergence, Many disguises of graph limits, Convergence of spectra, Convergence in norm, First applications

12. Convergence from the right
Homomorphisms to the right and multicuts, The overlay functional, Right-convergent graphon sequences, Right-convergent graph sequences

13. On the structure of graphons
The general form of a graphon, Weak isomorphism III, Pure kernels, The topology of a graphon, Symmetries of graphons

14. The space of graphons
Norms defined by graphs, Other norms on the kernel space, Closures of graph properties, Graphon varieties, Random graphons, Exponential random graph models

15. Algorithms for large graphs and graphons
Parameter estimation, Distinguishing graph properties, Property testing, Computable structures

16. Extremal theory of dense graphs
Nonnegativity of quantum graphs and reflection positivity, Variational calculus of graphons, Densities of complete graphs, The classical theory of extremal graphs, Local vs. global optima, Deciding inequalities between subgraph densities, Which graphs are extremal?

17. Multigraphs and decorated graphs
Compact decorated graphs, Multigraphs with unbounded edge multiplicities

Part 4. Limits of bounded degree graphs

18. Graphings
Borel graphs, Measure preserving graphs, Random rooted graphs, Subgraph densities in graphings, Local equivalence, Graphings and groups

19. Convergence of bounded degree graphs
Local convergence and limit, Local-global convergence

20. Right convergence of bounded degree graphs
Random homomorphisms to the right, Convergence from the right

21. On the structure of graphings
Hyperfiniteness, Homogeneous decomposition

22. Algorithms for bounded degree graphs
Estimable parameters, Testable properties, Computable structures

Part 5. Extensions: a brief survey

23. Other combinatorial structures
Sparse (but not very sparse) graphs, Edge-coloring models, Hypergraphs, Categories, And more

Appendix
M¨obius functions, The Tutte polynomial, Some background in probability and measure theory, Moments and the moment problem, Ultraproduct and ultralimit, Vapnik–Chervonenkis dimension, Nonnegative polynomials, Categories

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