[论文解读] Degree-degree dependencies in random graphs with heavy-tailed degrees

Remco van der Hofstad, Nelly Litvak|arXiv (Cornell University)|Jan 1, 2014

Complex Network Analysis Techniques参考文献 42被引用 21

一句话总结

本文表明，在具有重尾度分布的无标度网络中，度-度依赖关系的皮尔逊相关系数在大规模网络中不可靠，即使存在强烈的负相关性，该系数也常收敛于零或非负值。本文提出使用斯皮尔曼等级相关系数（rho）作为稳健的替代方法，并证明其在一般网络模型下收敛于有意义的极限，从而实现跨网络规模的一致性混合模式分析。

ABSTRACT

Mixing patterns in large self-organizing networks, such as the Internet, the World Wide Web, social and biological networks are often characterized by degree-degree dependencies between neighbouring nodes. In assortative networks, the degree-degree dependencies are positive (nodes with similar degrees tend to connect to each other), while in disassortative networks, these dependencies are negative. One of the problems with the commonly used Pearson correlation coecient, also known as the assortativity coecient is that its magnitude decreases with the network size in disassortative networks. This makes it impossible to compare mixing patterns, for example, in two web crawls of dierent sizes. As an alternative, we have recently suggested to use rank correlation measures, such as Spearman’s rho. Numerical experiments have conrmed that Spearman’s rho produces consistent values in graphs of dierent sizes but similar structure, and it is able to reveal strong (positive or negative) dependencies in large graphs. In this paper we analytically investigate degree-degree dependencies for scale-free graph sequences. In order to demonstrate the ill behaviour of the Pearson’s correlation coecient, we rst study a simple model of two heavy-tailed highly correlated random variables X and Y , and show that the sample correlation coecient converges in distribution either to a proper random variable on [ 1; 1], or to zero, and the limit is non-negative a.s. if X;Y 0. We next adapt these results to the degree-degree dependencies in networks as described by the Pearson correlation coecient, and show that it is non-negative in the large graph limit when the asymptotic degree distribution has an innite third moment. Furthermore, we provide examples where the Pearson’s correlation coecient converges to zero in a network with strong negative degree-degree dependencies, and another example where this coecient converges in distribution to a random variable. We suggest the alternative degree-degree dependency measure, based on Spearman’s rho, and prove that this statistical estimator converges to an appropriate limit under quite general conditions. These conditions are proved to hold in common network models, such as the conguration model and the preferential attachment model. We conclude that rank correlations provide a suitable and informative method for uncovering network mixing patterns.

研究动机与目标

• 识别皮尔逊相关系数在测量大规模重尾网络中度-度依赖关系时的局限性。
• 证明即使在网络中存在强烈负度相关性时，皮尔逊相关系数仍可能收敛于零或非负值。
• 提出斯皮尔曼等级相关系数（rho）作为测量无标度网络中度-度依赖关系的更可靠替代方法。
• 证明在一般网络模型（如配置模型和优先连接模型）下，斯皮尔曼等级相关系数（rho）收敛于稳定极限。
• 通过使用基于秩的相关性度量，实现跨不同规模网络的混合模式一致性比较。

提出的方法

• 分析两个重尾且高度相关的随机变量 X 和 Y 的样本皮尔逊相关系数的渐近行为。
• 将双变量重尾分布的理论结果适配至随机图中的度-度依赖关系。
• 证明当度分布具有无限三阶矩时，皮尔逊相关系数在分布上收敛于非负随机变量或零。
• 引入斯皮尔曼等级相关系数（rho）作为测量度-度依赖关系的基于秩的替代方法。
• 证明在一般条件下（包括配置模型和优先连接模型）下，斯皮尔曼等级相关系数（rho）收敛于明确定义的极限。
• 利用理论收敛结果，建立基于秩的相关性在大规模网络分析中的一致性与可靠性。

实验结果

研究问题

• RQ1 为何皮尔逊相关系数在大规模无标度网络中无法检测到强烈的负度-度依赖关系？
• RQ2 在具有无限三阶矩的网络中，皮尔逊相关系数在何种条件下收敛于零或非负值？
• RQ3 斯皮尔曼等级相关系数（rho）能否在不同规模的网络中一致估计度-度依赖关系？
• RQ4 何种理论条件可确保斯皮尔曼等级相关系数（rho）在随机图模型中收敛于有意义的极限？
• RQ5 在具有重尾度分布的网络中，皮尔逊相关系数与斯皮尔曼等级相关系数（rho）的收敛特性有何不同？

主要发现

• 在具有无限三阶矩的大规模网络中，度-度依赖关系的皮尔逊相关系数在分布上收敛于非负随机变量或零，即使存在强烈的负相关性。
• 在具有重尾度分布且存在强烈负混合的网络中，皮尔逊相关系数可能收敛于零，导致其无法有效检测此类依赖关系。
• 斯皮尔曼等级相关系数（rho）在一般条件下收敛于稳定极限，包括在配置模型和优先连接模型中。
• 斯皮尔曼等级相关系数（rho）的收敛性在不同网络规模下均具有鲁棒性，使其适用于混合模式的一致性比较。
• 理论分析证实，在大规模重尾网络中，基于秩的相关性度量比皮尔逊相关系数更具信息量且更可靠。
• 本研究确立了斯皮尔曼等级相关系数（rho）可作为网络混合模式的一致且可解释的度量，即使皮尔逊相关系数失效。