The identification of the set of k most central nodes of a graph, or centrality maximization, is a key task in network analysis, with various applications ranging from finding communities in social and biological networks to understanding which seed nodes are important to diffuse information in a graph. As the exact computation of centrality measures does not scale to modern-sized networks, the most practical solution is to resort to rigorous, but efficiently computable, randomized approximations. In this work we present CentRA, the first algorithm based on progressive sampling to compute high-quality approximations of the set of k most central nodes. CentRA is based on a novel approach to efficiently estimate Monte Carlo Rademacher Averages, a powerful tool from statistical learning theory to compute sharp data-dependent approximation bounds. Then, we study the sample complexity of centrality maximization using the VC-dimension, a key concept from statistical learning theory. We show that the number of random samples required to compute high-quality approximations scales with finer characteristics of the graph, such as its vertex diameter, or of the centrality of interest, significantly improving looser bounds derived from standard techniques. We apply CentRA to analyze large real-world networks, showing that it significantly outperforms the state-of-the-art approximation algorithm in terms of number of samples, running times, and accuracy.

翻译：识别图中k个最中心节点的集合，即中心性最大化，是网络分析中的关键任务，具有从发现社交和生物网络中的社区到了解哪些种子节点对图中信息扩散至关重要等多种应用。由于中心性度量的精确计算无法扩展到现代规模网络，最实用的解决方案是采用严谨但可高效计算的随机近似方法。本文提出CentRA，这是首个基于渐进采样以计算k个最中心节点集合的高质量近似值的算法。CentRA基于一种高效估计蒙特卡洛Rademacher平均的新方法，这是统计学习理论中用于计算精确数据依赖近似界的强大工具。随后，我们利用统计学习理论的关键概念VC维研究了中心性最大化的样本复杂度。研究表明，计算高质量近似值所需的随机样本数量与图的更精细特征（如顶点直径）或所关注的中心性指标成比例，显著改进了标准技术推导出的宽松界限。我们将CentRA应用于分析大型真实世界网络，结果表明其在样本数量、运行时间和准确性方面均显著优于最先进的近似算法。