[论文解读] Counting Motifs with Graph Sampling

Jason M. Klusowski, Yihong Wu|arXiv (Cornell University)|Feb 21, 2018

Complex Network Analysis Techniques参考文献 29被引用 3

一句话总结

本文研究在大规模网络中通过子图采样和邻域采样对基序计数（诱导子图）进行统计估计。它建立了以乘法误差估计基序计数的最优采样比例，表明邻域采样显著优于子图采样，且结果仅依赖于基序大小和图度数，而不依赖于图的拓扑结构。

ABSTRACT

Applied researchers often construct a network from a random sample of nodes in order to infer properties of the parent network. Two of the most widely used sampling schemes are subgraph sampling, where we sample each vertex independently with probability $p$ and observe the subgraph induced by the sampled vertices, and neighborhood sampling, where we additionally observe the edges between the sampled vertices and their neighbors. In this paper, we study the problem of estimating the number of motifs as induced subgraphs under both models from a statistical perspective. We show that: for any connected $h$ on $k$ vertices, to estimate $s=\mathsf{s}(h,G)$, the number of copies of $h$ in the parent graph $G$ of maximum degree $d$, with a multiplicative error of $ε$, (a) For subgraph sampling, the optimal sampling ratio $p$ is $Θ_{k}(\max\{ (sε^2)^{-\frac{1}{k}}, \; \frac{d^{k-1}}{sε^{2}} \})$, achieved by Horvitz-Thompson type of estimators. (b) For neighborhood sampling, we propose a family of estimators, encompassing and outperforming the Horvitz-Thompson estimator and achieving the sampling ratio $O_{k}(\min\{ (\frac{d}{sε^2})^{\frac{1}{k-1}}, \; \sqrt{\frac{d^{k-2}}{sε^2}}\})$. This is shown to be optimal for all motifs with at most $4$ vertices and cliques of all sizes. The matching minimax lower bounds are established using certain algebraic properties of subgraph counts. These results quantify how much more informative neighborhood sampling is than subgraph sampling, as empirically verified by experiments on both synthetic and real-world data. We also address the issue of adaptation to the unknown maximum degree, and study specific problems for parent graphs with additional structures, e.g., trees or planar graphs.

研究动机与目标

• 解决在仅获得采样子图时，大规模网络中基序计数的统计估计挑战。
• 在实际采样约束下，比较子图采样与邻域采样在基序计数中的效率。
• 推导出对任意连通基序（大小为k）最小化估计误差的最优采样比例。
• 在两种采样模型下，为基序计数开发普遍最优的估计器——特别是Horvitz-Thompson型估计器。
• 建立极小化最大风险下界，并证明所提估计器对小基序和团的最优性。

提出的方法

• 为子图采样提出一种Horvitz-Thompson型估计器，证明其在所有连通基序中均为普遍最优。
• 引入一组邻域采样的估计器，其为广义形式且优于Horvitz-Thompson估计器。
• 利用子图计数的代数性质与极小化最大风险分析，推导最优采样比例。
• 通过组合与代数技术建立匹配的极小化最大下界，以证明所提采样策略的最优性。
• 分析在最大度数d未知时的适应性，采用数据驱动的采样比例选择方法。
• 考虑特殊图结构（如树、平面图）以在结构约束下进一步提升采样效率。

实验结果

研究问题

• RQ1 子图采样中，为以乘法误差ε估计基序计数，最优采样比例p是多少？
• RQ2 与子图采样相比，邻域采样在基序计数中如何提升估计效率？
• RQ3 能否在两种采样模型下，为所有连通基序构造普遍最优的估计器？
• RQ4 子图采样与邻域采样下，基序计数的极小化最大下界是什么？
• RQ5 当父图的最大度数d未知时，如何调整采样比例？

主要发现

• 对于子图采样，最优采样比例为Θk(max{(sε²)^(-1/k), d^{k-1}/(sε²)})，与基序拓扑无关。
• Horvitz-Thompson估计器在子图采样下对所有连通基序均为普遍最优。
• 对于邻域采样，所提估计器实现的采样比例为Ok(min{(d/(sε²))^{1/(k-1)}, √(d^{k-2}/(sε²))})，同样与基序结构无关。
• 邻域采样估计器对最多含4个顶点的基序以及任意大小的团均为最优。
• 极小化最大下界与上界匹配，证明了所提采样策略的最优性。
• 在合成数据与真实世界数据上的实验表明，邻域采样显著优于子图采样，估计精度更高。