Kronecker图：一种网络建模方法
Kronecker Graphs: An Approach to Modeling Networks
摘要：
我们如何用一个数学上易于处理的模型来对网络进行建模，以便对网络属性进行严格的分析？网络表现出一长串令人惊讶的特性：度分布的厚尾;小直径;随着时间的推移，密度和直径收缩。大多数现有的网络模型要么无法匹配上述几个属性，要么数学分析复杂，或者两者兼而有之。在本文中，我们提出了一个生成模型的网络，这是数学上易于处理，并可以生成具有上述属性的网络。我们的主要思想是使用克罗内克积来生成我们称为“克罗内克图”的图。首先，我们证明了Kronecker图自然服从共同的网络性质。我们还提供了经验证据表明，克罗内克图可以有效地模拟真实网络的结构。然后，我们提出了KronFit，一个快速和可扩展的算法，用于拟合Kronecker图生成模型到大型真实网络。一个简单的拟合方法将花费超指数的时间。相比之下，KronFit通过利用Kronecker矩阵乘法的结构和使用统计模拟技术，需要线性时间。在大型真实和合成网络上的实验表明，KronFit找到了准确的参数，这些参数确实非常好地模拟了目标网络的属性。一旦拟合，模型参数可以用于获得关于网络结构的见解，并且所得到的合成图可以用于空模型、匿名化、外推和图总结。
英文摘要：
How can we model networks with a mathematically tractable model that allows for rigorous analysis of network properties? Networks exhibit a long list of surprising properties: heavy tails for the degree distribution; small diameters; and densification and shrinking diameters over time. Most present network models either fail to match several of the above properties, are complicated to analyze mathematically, or both. In this paper we propose a generative model for networks that is both mathematically tractable and can generate networks that have the above mentioned properties. Our main idea is to use the Kronecker product to generate graphs that we refer to as "Kronecker graphs". First, we prove that Kronecker graphs naturally obey common network properties. We also provide empirical evidence showing that Kronecker graphs can effectively model the structure of real networks. We then present KronFit, a fast and scalable algorithm for fitting the Kronecker graph generation model to large real networks. A naive approach to fitting would take super- exponential time. In contrast, KronFit takes linear time, by exploiting the structure of Kronecker matrix multiplication and by using statistical simulation techniques. Experiments on large real and synthetic networks show that KronFit finds accurate parameters that indeed very well mimic the properties of target networks. Once fitted, the model parameters can be used to gain insights about the network structure, and the resulting synthetic graphs can be used for null- models, anonymization, extrapolations, and graph summarization.
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。
