基于无分布随机图分析的鲁棒子图计数
Robust subgraph counting with distribution-free random graph analysis
摘要：
子图，如团，环和星形成了现实世界网络拓扑结构中的关键连接。随机图模型提供了对某些子图出现频率的估计，这反过来又可以在现实世界的网络中进行测试。然而，这些子图的数量主要取决于假设的度分布。将度分布拟合到网络数据是一个挑战，特别是对于具有幂律度的无标度网络。在本文中，我们开发了鲁棒的子图计数，它不依赖于整个度分布，而只依赖于平均值和平均绝对偏差（MAD），这是大多数真实网络很容易获得的汇总统计数据。通过解决一个优化问题，我们为子图计数、所有可能的子图以及具有相同均值和MAD的度分布的所有网络提供了严格的（最尖锐的）边界。我们确定的极值随机图，达到紧界的图与一个特定的三点度分布。我们利用边界来获得鲁棒的缩放定律，子图的数量如何作为网络大小的函数增长。缩放定律表明稀疏幂律网络在子图数方面不是最极端的网络，但稠密幂律网络是。强大的界限也被证明是几个现实世界的数据集。
英文摘要：
Subgraphs such as cliques, loops and stars form crucial connections in the topologies of real-world networks. Random graph models provide estimates for how often certain subgraphs appear, which in turn can be tested against real-world networks. These subgraph counts, however, crucially depend on the assumed degree distribution. Fitting a degree distribution to network data is challenging, in particular for scale-free networks with power-law degrees. In this paper we develop robust subgraph counts that do not depend on the entire degree distribution, but only on the mean and mean absolute deviation (MAD), summary statistics that are easy to obtain for most real-world networks. By solving an optimization problem, we provide tight (the sharpest possible) bounds for the subgraph counts, for all possible subgraphs, and for all networks with degree distributions that share the same mean and MAD. We identify the extremal random graph that attains the tight bounds as the graph with a specific three-point degree distribution. We leverage the bounds to obtain robust scaling laws for how the numbers of subgraphs grow as function of the network size. The scaling laws indicate that sparse power-law networks are not the most extreme networks in terms of subgraph counts, but dense power-law networks are. The robust bounds are also shown to hold for several real-world data sets.
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