[论文解读] Refining Approximating Betweenness Centrality Based on Samplings

Shiyu Ji, Zenghui Yan|arXiv (Cornell University)|Aug 16, 2016

Complex Network Analysis Techniques参考文献 17被引用 6

一句话总结

本文重新审视了Bader基于采样的介数中心性（BC）近似算法，识别出其界中存在的理论缺陷，并提出一种改进的采样方法，实现了更紧致的概率保证。该文引入了一种基于顶点对的自适应采样算法，相较于以往工作，实现了更低的近似因子、更高的成功概率以及更少的采样次数，在真实网络上的实验验证显示，性能提升最高达7%的因子差异。

ABSTRACT

Betweenness Centrality (BC) is an important measure used widely in complex network analysis, such as social network, web page search, etc. Computing the exact BC values is highly time consuming. Currently the fastest exact BC determining algorithm is given by Brandes, taking $O(nm)$ time for unweighted graphs and $O(nm+n^2\log n)$ time for weighted graphs, where $n$ is the number of vertices and $m$ is the number of edges in the graph. Due to the extreme difficulty of reducing the time complexity of exact BC determining problem, many researchers have considered the possibility of any satisfactory BC approximation algorithms, especially those based on samplings. Bader et al. give the currently best BC approximation algorithm, with a high probability to successfully estimate the BC of one vertex within a factor of $1/\varepsilon$ using $\varepsilon t$ samples, where $t$ is the ratio between $n^2$ and the BC value of the vertex. However, some of the algorithmic parameters in Bader's work are not yet tightly bounded, leaving some space to improve their algorithm. In this project, we revisit Bader's algorithm, give more tight bounds on the estimations, and improve the performance of the approximation algorithm, i.e., fewer samples and lower factors. On the other hand, Riondato proposes a non-adaptive BC approximation method with samplings on shortest paths. We investigate the possibility to give an adaptive algorithm with samplings on vertices. With rigorous reasoning, we show that this algorithm can also give bounded approximation factors and number of samplings. To evaluate our results, we also conduct extensive experiments using real-world datasets, and find that both our algorithms can achieve acceptable performance. We verify that our work can improve the current BC approximation algorithm with better parameters, i.e., higher successful probabilities, lower factors and fewer samples.

研究动机与目标

• 为解决Bader的BC近似算法中的理论缺口，特别是概率界缺陷和未经证明的引理。
• 通过减少所需采样次数和近似因子，提升BC近似算法的性能。
• 提出一种基于顶点对的新型自适应采样策略，确保近似误差有界。
• 通过真实世界图数据集对新算法进行严格分析与验证。
• 建立比现有方法更紧致的近似因子与采样复杂度的理论界。

提出的方法

• 重新审视Bader的算法，并通过构建新的概率模型，修正其中存在缺陷的概率论证，特别是引理3中的问题。
• 提出一种改进的Bader式算法，结合Bader与Riondato的采样策略，采用混合方法。
• 引入一种自适应采样方法，根据动态阈值选择顶点对，以提升估计精度。
• 推导理论界，表明近似因子被限制在 $ b = t^{1/3}/S $ 以内，其中 $ S $ 为采样次数，$ t $ 为 $ n^2 $ 与BC值的比值。
• 在采样循环中采用基于阈值的终止条件，其中更高的 $ c $ 值允许更多采样并获得更紧的界。
• 通过在真实世界图上的大规模实验验证性能，比较因子差异与采样次数。

实验结果

研究问题

• RQ1 Bader的BC近似算法中的理论界能否被进一步收紧，以提升精度并减少采样需求？
• RQ2 Bader原始证明中存在缺陷的概率推理有何影响，应如何修正？
• RQ3 基于顶点对的自适应采样策略是否能以更少的采样次数实现比均匀采样更低的近似因子？
• RQ4 近似因子的倒数与采样次数之间是否存在理论关系？
• RQ5 所提出的算法在真实世界稀疏网络上的表现相较于现有方法如何？

主要发现

• 在测试图中，改进后的Bader算法即使采样数少于800次，因子差异也始终低于7%。
• 因子差异的倒数与采样次数近似呈线性关系，支持理论预测。
• 自适应对采样算法的因子差异低于5%，在精度上优于改进后的Bader方法。
• 理论分析证实，近似因子被限制在 $ b = t^{1/3}/S $ 以内，且界比Bader原始工作更紧致。
• 采样次数几乎随阈值参数 $ c $ 线性增长，表明行为具有可预测性与可扩展性。
• 实验结果证实，两种算法在更少采样次数下均实现了更高的成功概率与更低的因子，优于以往基于采样的BC近似方法。