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.

本文中介绍了一种名为 Kronecker graphs 的生成模型，可以截尾分布、小直径、密度增加和直径缩小等特性自动生成网络。使用 Kronecker 矩阵乘法结构和统计模拟技术，提出了一种快速、可伸缩的算法 KronFit，可用于拟合真实网络的 Kronecker 图形成模型参数，提取网络结构信息和生成综合图形。