Over the years, quantifying the similarity of nodes has been a hot topic in network science, yet little has been known about the distribution of node-similarity. In this paper, we consider a typical measure of node-similarity called the common neighbor based similarity (CNS). By means of the generating function, we propose a general framework for calculating the CNS distributions of node sets in various networks. Particularly, we show that for the Erdös-Rényi random network, the CNS distribution of node sets of any size obeys the Poisson law. Furthermore, we connect the node-similarity distribution to the link prediction problem, and derive analytical solutions for two key evaluation metrics: i) precision and ii) area under the receiver operating characteristic curve (AUC). We also use the similarity distributions to optimize link prediction by i) deriving the expected prediction accuracy of similarity scores and ii) providing the optimal prediction priority of unconnected node pairs. Simulation results confirm our theoretical findings and also validate the proposed tools in evaluating and optimizing link prediction.

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复杂网络的节点相似度分布及其在链路预测中的应用

多年来，量化节点的相似度一直是网络科学的热门话题，但对节点相似度的分布知之甚少。在本文中，我们考虑了一种典型的节点相似性度量，称为基于公共邻居的相似性 (CNS)。通过生成函数，我们提出了一个用于计算各种网络中节点集的 CNS 分布的通用框架。特别是，我们表明对于 Erdös-Rényi 随机网络，任何大小的节点集的 CNS 分布都遵循泊松定律。此外，我们将节点相似性分布与链路预测问题联系起来，并推导出两个关键评估指标的解析解：i) 精度和 ii) 接收器操作特征曲线 (AUC) 下的面积。我们还使用相似性分布来优化链接预测，方法是 i) 得出相似性分数的预期预测准确度和 ii) 提供未连接节点对的最佳预测优先级。仿真结果证实了我们的理论发现，并验证了所提出的评估和优化链路预测的工具。