标题： 几何和无标度网络中的 assortativity 显示英文标题

标题： Assortativity in geometric and scale-free networks

摘要： 网络的同配行为是指相似（或不相似）节点相互连接的倾向。 这种倾向可以影响网络的各种特性，例如其鲁棒性或传播过程的动力学特性。 在本文中，我们研究了现实世界网络以及基于潜在空间的具有重尾度分布的几种生成模型中的度同配性。 特别是，我们研究了Chung-Lu图和几何非均匀随机图（GIRGs）。 以往关于同配性的研究主要集中在使用皮尔逊同配系数来测量现实世界网络中的度同配性，尽管对此系数存在保留意见。 我们通过数学证明严格确认了这些保留意见，证明皮尔逊同配系数在具有足够重尾度分布的任何网络中都不测量同配性，而这是现实世界网络的典型特征。 此外，我们发现其他单值同配系数也不足以捕捉节点的布线偏好，因为这些偏好通常因节点度数而有很大差异。 因此，我们采用了一种更细致的方法，分析了连接节点的广泛条件和联合权重及度分布，既在现实世界网络中进行数值分析，也在生成图模型中进行数学分析。 我们提供了几种可视化结果的方法。 我们表明生成模型是同配性中立的，而许多现实世界网络并非如此。 因此，我们还提出了一种GIRG模型的扩展，该模型保留了由度分布和潜在空间引起的多种理想特性，但也表现出可调节的同配性。 我们对产生的模型进行了数学分析，并对其同配性进行了细致的量化。 显示英文摘要

摘要： The assortative behavior of a network is the tendency of similar (or dissimilar) nodes to connect to each other. This tendency can have an influence on various properties of the network, such as its robustness or the dynamics of spreading processes. In this paper, we study degree assortativity both in real-world networks and in several generative models for networks with heavy-tailed degree distribution based on latent spaces. In particular, we study Chung-Lu Graphs and Geometric Inhomogeneous Random Graphs (GIRGs). Previous research on assortativity has primarily focused on measuring the degree assortativity in real-world networks using the Pearson assortativity coefficient, despite reservations against this coefficient. We rigorously confirm these reservations by mathematically proving that the Pearson assortativity coefficient does not measure assortativity in any network with sufficiently heavy-tailed degree distributions, which is typical for real-world networks. Moreover, we find that other single-valued assortativity coefficients also do not sufficiently capture the wiring preferences of nodes, which often vary greatly by node degree. We therefore take a more fine-grained approach, analyzing a wide range of conditional and joint weight and degree distributions of connected nodes, both numerically in real-world networks and mathematically in the generative graph models. We provide several methods of visualizing the results. We show that the generative models are assortativity-neutral, while many real-world networks are not. Therefore, we also propose an extension of the GIRG model which retains the manifold desirable properties induced by the degree distribution and the latent space, but also exhibits tunable assortativity. We analyze the resulting model mathematically, and give a fine-grained quantification of its assortativity.