大型网络和图极限(影印版)

世界上许多有趣的结构 和现象可以用网络来描述。 发展大型网络的数学理论是 重要的挑战。本书描述了最 近十年出现的新方法——图 极限理论。该理论与研究大 型网络的其他方法，如计算 机科学中的“性质检验”和图 论中的正则划分，有着丰富 的联系。它在极值图论中有 一些应用，包括非常普遍的 问题的确切公式和部分答案 ，例如图极限理论中哪些问 题是可判定的。它还与数学 的其他领域（经典和非经典 的，如概率论、测度论、张 量代数和半正定优化）有着 不易察觉的联系。

Preface Part 1.Large graphs: an informal introduction Chapter 1.Very large networks 1.1 Huge networks everywhere 1.2 What to ask about them 1.3 How to obtain information about them 1.4 How to model them 1.5 How to approximate them 1.6 How to run algorithms on them 1.7 Bounded degree graphs Chapter 2.Large graphs in mathematics and physics 2.1 Extremal graph theory 2.2 Statistical physics Part 2.The algebra of graph homomorphisms Chapter 3.Notation and terminology 3.1 Basic notation 3.2 Graph theory 3.3 Operations on graphs Chapter 4.Graph parameters and connection matrices 4.1 Graph parameters and graph properties 4.2 Connection matrices 4.3 Finite connection rank Chapter 5.Graph homomorphisms 5.1 Existence of homomorphisms 5.2 Homomorphism numbers 5.3 What hom functions can express 5.4 Homomorphism and isomorphism 5.5 Independence of homomorphism functions 5.6 Characterizing homomorphism numbers 5.7 The structure of the homomorphism set Chapter 6.Graph algebras and homomorphism functions 6.1 Algebras of quantum graphs 6.2 Reflection positivity 6.3 Contractors and connectors 6.4 Algebras for homomorphism functions 6.5 Computing parameters with finite connection rank 6.6 The polynomial method Part 3.Limits of dense graph sequences Chapter 7.Kernels and graphons 7.1 Kernels, graphons and stepfunctions 7.2 Generalizing homomorphisms 7.3 Weak isomorphism I 7.4 Sums and products 7.5 Kernel operators Chapter 8.The cut distance 8.1 The cut distance of graphs 8.2 Cut norm and cut distance of kernels 8.3 Weak and L1-topologies Chapter 9.Szemeredi partitions 9.1 Regularity Lemma for graphs