arXiv 0812.26562008-12math.PRmath.CO

稀疏图：度量和随机模型

Sparse graphs: metrics and random models

Bela Bollobas, Oliver Riordan

备注：: 44 pages, 1 figure. This is a companion paper to arXiv:0708.1919, consisting of an updated version of part of the original version (arXiv:0708.1919v1), which has been split into two papers. Since v1, references updated and other very minor changes. To appear in Random Structures and Algorithms

摘要：

最近，Bollobás，Janson和Riordan引入了一类随机图模型，产生具有$n$顶点和$Θ(n)$边的非齐次图，其分布由核表征，即，对称可测函数$\ka:[0,1]^2 \to [0,\infty)$。为了理解这些模型，我们应该知道不同的内核$\ka$何时产生“相似”的图，并且，给定一个真实世界的网络，它与从给定内核$\ka$导出的典型图$G(n,\ka)$有多“相似”。Borgs，Chayes，Lovász，Sós，Szegedy和Vesztergombi最近的结果回答了具有$Θ(n^2)$边的稠密图的类似问题，他们证明了图上的几个自然度量是等价的，而且任何图序列在每个度量中都收敛到一个图子，即，在$[0,1]$中取值的内核。这些结果可能推广到具有$o(n^2)$但具有$ω(n)$边的图中，在一篇配套论文[arXiv：0708.1919]中进行了讨论;在这里，我们只关注具有$Θ(n)$边的图，结果证明这更难处理。许多新的现象出现了，并且有许多合理的指标需要考虑;这些指标中的许多指标建议新的随机图模型，反之亦然。

英文摘要：

Recently, Bollobás, Janson and Riordan introduced a family of random graph models producing inhomogeneous graphs with $n$ vertices and $Θ(n)$ edges whose distribution is characterized by a kernel, i.e., a symmetric measurable function $\ka:[0,1]^2 \to [0,\infty)$. To understand these models, we should like to know when different kernels $\ka$ give rise to `similar' graphs, and, given a real-world network, how `similar' is it to a typical graph $G(n,\ka)$ derived from a given kernel $\ka$. The analogous questions for dense graphs, with $Θ(n^2)$ edges, are answered by recent results of Borgs, Chayes, Lovász, Sós, Szegedy and Vesztergombi, who showed that several natural metrics on graphs are equivalent, and moreover that any sequence of graphs converges in each metric to a graphon, i.e., a kernel taking values in $[0,1]$. Possible generalizations of these results to graphs with $o(n^2)$ but $ω(n)$ edges are discussed in a companion paper [arXiv:0708.1919]; here we focus only on graphs with $Θ(n)$ edges, which turn out to be much harder to handle. Many new phenomena occur, and there are a host of plausible metrics to consider; many of these metrics suggest new random graph models, and vice versa.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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