稠密图的收敛序列I：子图频率、度量性质及检验
Convergent Sequences of Dense Graphs I: Subgraph Frequencies, Metric Properties and Testing
摘要：
我们考虑图的序列，并定义与这些序列相关的各种收敛概念：“左收敛”定义为从小图到序列图的同态密度，“右收敛”定义为从序列图到小图的同态密度;以及在适当定义的度量中收敛。在本系列的第一部分中，我们证明了左收敛等价于度量收敛，无论是对于单图，还是对于具有节点权和边权的图。这里的一个主要步骤是引入一个切割距离比较图，不一定是相同的大小。我们还展示了这些收敛的概念如何提供自然配方的Szemeredi分区，大型图形的采样和测试。
英文摘要：
We consider sequences of graphs and define various notions of convergence related to these sequences: ``left convergence'' defined in terms of the densities of homomorphisms from small graphs into the graphs of the sequence, and ``right convergence'' defined in terms of the densities of homomorphisms from the graphs of the sequence into small graphs; and convergence in a suitably defined metric. In Part I of this series, we show that left convergence is equivalent to convergence in metric, both for simple graphs, and for graphs with nodeweights and edgeweights. One of the main steps here is the introduction of a cut-distance comparing graphs, not necessarily of the same size. We also show how these notions of convergence provide natural formulations of Szemeredi partitions, sampling and testing of large graphs.
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