捕捉头部、尾部和两者之间的一切：度分布的流算法
Catching the head, tail, and everything in between: a streaming algorithm for the degree distribution
摘要：
度分布是真实世界图中最基本的图形属性之一。在许多领域中已经广泛观察到，图通常具有尾部或无标度分布。虽然平均度通常很小，但方差很高，并且在所有尺度下都有具有度的顶点。我们专注于一个大的流图的度分布的近似问题，与小存储。我们设计了一个算法headtail，其主要的新颖性是一个新的估计罕见度截断几何随机变量。我们给出了一个数学分析的头尾，并表明它具有良好的性能在实践中。我们可以处理数百万条边的流，存储量不到1%，并对度分布中的所有尺度获得非常准确的近似。我们还引入了一个新的概念之间的相对Hausdorff距离尾直方图。现有的分布之间的距离的概念是不合适的，因为它们忽略了尾部的罕见度。相对豪斯多夫距离测量所有尺度的偏差，是比较度分布的更合适的距离。通过跟踪这一新的措施，我们能够给出强有力的实证证据的收敛头尾。
英文摘要：
The degree distribution is one of the most fundamental graph properties of interest for real-world graphs. It has been widely observed in numerous domains that graphs typically have a tailed or scale-free degree distribution. While the average degree is usually quite small, the variance is quite high and there are vertices with degrees at all scales. We focus on the problem of approximating the degree distribution of a large streaming graph, with small storage. We design an algorithm headtail, whose main novelty is a new estimator of infrequent degrees using truncated geometric random variables. We give a mathematical analysis of headtail and show that it has excellent behavior in practice. We can process streams will millions of edges with storage less than 1% and get extremely accurate approximations for all scales in the degree distribution. We also introduce a new notion of Relative Hausdorff distance between tailed histograms. Existing notions of distances between distributions are not suitable, since they ignore infrequent degrees in the tail. The Relative Hausdorff distance measures deviations at all scales, and is a more suitable distance for comparing degree distributions. By tracking this new measure, we are able to give strong empirical evidence of the convergence of headtail.
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