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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

  • Read Online Large Networks and Graph Limits by Laszlo Lovasz
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