具有聚类的稀疏随机图
Sparse random graphs with clustering
摘要：
在2007年，我们引入了一个一般模型的稀疏随机图之间的独立的边缘。本文的目的是提出一个扩展的边缘是远离独立的这个模型，并证明了几个结果，这种扩展。其基本思想是通过添加边和其他小图来构造随机图。换句话说，我们首先构造一个具有独立超边的非齐次随机超图，然后用一个（可能是完全的）图来替换每个超边。虽然足够灵活，可以生成边之间具有显著相关性的图，但该模型在数学上仍然是易于处理的。事实上，我们找到了一个巨大的组件出现在充分的一般性的临界点，在一定的积分算子的规范，并涉及到一定的（非泊松）多类型分支过程的生存概率的巨大的组件的大小。虽然我们的主要重点是相变，我们也研究了度分布和小的子图的数量。我们用一个简单的特例来说明这个模型，这个特例产生了幂律度序列的图，它具有广泛的度指数和聚类系数。
英文摘要：
In 2007 we introduced a general model of sparse random graphs with independence between the edges. The aim of this paper is to present an extension of this model in which the edges are far from independent, and to prove several results about this extension. The basic idea is to construct the random graph by adding not only edges but also other small graphs. In other words, we first construct an inhomogeneous random hypergraph with independent hyperedges, and then replace each hyperedge by a (perhaps complete) graph. Although flexible enough to produce graphs with significant dependence between edges, this model is nonetheless mathematically tractable. Indeed, we find the critical point where a giant component emerges in full generality, in terms of the norm of a certain integral operator, and relate the size of the giant component to the survival probability of a certain (non-Poisson) multi-type branching process. While our main focus is the phase transition, we also study the degree distribution and the numbers of small subgraphs. We illustrate the model with a simple special case that produces graphs with power-law degree sequences with a wide range of degree exponents and clustering coefficients.
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。
