随机图与复杂网络中聚类与度指数的分析

Extremal clustering and cluster counting for spatial random fields

We consider a stationary random field indexed by an increasing sequence of subsets of $\mathbb{Z}^d$ obeying a very broad geometrical assumption on how the sequence expands. Under certain mixing and local conditions, we show how the tail distribution of the individual variables relates to the tail behavior of the maximum of the field over the index sets in the limit as the index sets expand. Furthermore, in a framework where we let the increasing index sets be scalar multiplications of a fixed set $C$, potentially with different scalars in different directions, we use two cluster definitions t…

度分布和聚类可调的随机交图

Random intersection graphs with tunable degree distribution and clustering

随机交图是通过独立地分配给每个顶点的一个子集的给定集合和绘制两个顶点之间的边当且仅当它们各自的子集相交。本文提出了一个模型，其中每个顶点被赋予一个随机的权重，与较大的权重顶点更有可能被分配到大的子集。一个给定的顶点的程度的分布的特点，并示出依赖于顶点的重量。特别地，如果权重分布是幂律，则度分布也将是幂律。此外，在图中的聚类的渐近表达式推导。通过调整模型的参数，可以生成具有任意聚类、期望度和（在幂律情况下）尾指数的图。

Local Clustering Coefficient in Generalized Preferential Attachment Models

In this paper, we analyze the local clustering coefficient of preferential attachment models. A general approach to preferential attachment was introduced in earlier, where a wide class of models (PA-class) was defined in terms of constraints that are sufficient for the study of the degree distribution and the clustering coefficient. It was previously shown that the degree distribution in all models of the PA-class follows a power law. Also, the global clustering coefficient was analyzed and a lower bound for the average local clustering coefficient was obtained. We expand the results by analy…

稀疏随机交图的分类性与聚类

Assortativity and clustering of sparse random intersection graphs

我们考虑稀疏随机交图的性质，聚类系数不为零的节点数趋于无穷大。我们发现明确的渐近表达式的相关系数的程度相邻节点（称为的相关系数），预期数量的共同邻居的相邻节点，和预期程度的邻居的节点的一个给定的程度k。这些表达式是用渐近度分布来表示的，或者用定义底层随机图模型的参数来表示。

Correlation between clustering and degree in affiliation networks

We are interested in the probability that two randomly selected neighbors of a random vertex of degree (at least) $k$ are adjacent. We evaluate this probability for a power law random intersection graph, where each vertex is prescribed a collection of attributes and two vertices are adjacent whenever they share a common attribute. We show that the probability obeys the scaling $k^{-δ}$ as $k\to+\infty$. Our results are mathematically rigorous. The parameter $0\le δ\le 1$ is determined by the tail indices of power law random weights defining the links between vertices and attributes.

Scale-free percolation in continuum space: quenched degree and clustering coefficient

Spatial random graphs capture several important properties of real-world networks. We prove quenched results for the continuum space version of scale-free percolation introduced in [DW18]. This is an undirected inhomogeneous random graph whose vertices are given by a Poisson point process in $\mathbb R^d$. Each vertex is equipped with a random weight and the probability that two vertices are connected by an edge depends on their weights and on their distance. Under suitable conditions on the parameters of the model, we show that, for almost all realizations of the point process, the degree dis…

具有可调聚类、度相关和度分布的网络及其上的流行病

A network with tunable clustering, degree correlation and degree distribution, and an epidemic thereon

一个随机网络模型，它允许可调的，相当一般的形式的聚类，度相关性和度分布的定义。该模型是配置模型的扩展，其中存根（半边）配对形成网络。聚类是通过形成小的完全连通的子群，和正（负）度的相关性是通过连接一小部分的存根与存根的相似（不相似）的程度。在该网络上定义了一个SIR（Susceptible -> Infective -> Adverted）流行病模型.网络和流行病的渐近性质，作为总体规模趋于无穷大，推导出：度分布，度相关性和聚类系数，以及再生数$R_*$，一个大爆发的概率和相对大小这样的爆发。理论说明了Monte Carlo模拟和数值例子。主要研究结果是，聚类倾向于减少疾病的传播，当疾病接近阈值时，程度相关性的效果明显大于当疾病远高于阈值时的效果，当$R_*$刚好高于其门限值时，疾病传播随着程度相关性$ρ$而广泛增加，当$R_*$远高于1时，随着$ρ$而减少。

Sharpest possible clustering bounds using robust random graph analysis

Complex network theory crucially depends on the assumptions made about the degree distribution, while fitting degree distributions to network data is challenging, in particular for scale-free networks with power-law degrees. We present a robust assessment of complex networks that does not depend on the entire degree distribution, but only on its mean, range and dispersion: summary statistics that are easy to obtain for most real-world networks. By solving several semi-infinite linear programs, we obtain tight (the sharpest possible) bounds for correlation and clustering measures, for all netwo…

随机场中高值的聚类

Clustering of high values in random fields

作为极端随机场应用基础的渐近结果通常假设变量位于规则离散网格上，用$\mathbb{Z}^2$标识，并且它们满足平稳性和各向同性条件。在这里，我们扩展现有的理论，关于最大值和极值指数的渐近行为，非平稳和各向异性的随机场，定义在离散子集的$\mathbb{R}^2$。我们表明，在一个合适的coordinatewise长程依赖条件下，最大值可以被视为一个近似独立的序列的次最大值的最大值，虽然可能有很高的局部依赖性导致聚类的高值。在对高值局部路径行为的限制下，给出了空间极值指标的存在性和取值准则，该指标在确定簇的大小和量化高水平随机随机场之间的依赖强度方面起着关键作用。其中极值指数作为在文献中发现的公知的尾部依赖性措施的函数获得，从而导致该参数的简单估计方法。结果与非平稳高斯和1相关的随机场。对于后者，进行了模拟和估计研究。

可调聚类随机交图分支结构的一个注记

A note on the component structure in random intersection graphs with tunable clustering

我们研究了可调聚类的随机交图中的组件结构，并证明了平均度可以作为最大组件大小的相变阈值。也就是说，如果期望度小于1，则最大分量的大小为a.a.s.。对数级，但如果平均度数大于1，a.a.s.出现线性级的单个大分量，并且第二大分量的大小至多为对数级。

Scale-free network clustering in hyperbolic and other random graphs

Random graphs with power-law degrees can model scale-free networks as sparse topologies with strong degree heterogeneity. Mathematical analysis of such random graphs proved successful in explaining scale-free network properties such as resilience, navigability and small distances. We introduce a variational principle to explain how vertices tend to cluster in triangles as a function of their degrees. We apply the variational principle to the hyperbolic model that quickly gains popularity as a model for scale-free networks with latent geometries and clustering. We show that clustering in the hy…

低度多项式下界用于中等维度下的聚类

Low-degree Lower bounds for clustering in moderate dimension

本文研究了中等维度下聚类问题的低度多项式下界，提出了一种非谱算法，揭示了中等维度下聚类的计算限制。