In this paper we formally define the hierarchical clustering network problem (HCNP) as the problem to find a good hierarchical partition of a network. This new problem focuses on the dynamic process of the clustering rather than on the final picture of the clustering process. To address it, we introduce a new ierarchical clustering algorithm in networks, based on a new shortest path betweenness measure. To calculate it, the communication between each pair of nodes is weighed by he importance of the nodes that establish this communication. The weights or importance associated to each pair of nodes are calculated as the Shapley value of a game, named as the linear modularity game. This new measure, (the node-game shortest path betweenness measure), is used to obtain a hierarchical partition of the network by eliminating the link with the highest value. To evaluate the performance of our algorithm, we introduce several criteria that allow us to compare different dendrograms of a network from two point of view: modularity and homogeneity. Finally, we propose a faster algorithm based on a simplification of the node-game shortest path betweenness measure, whose order is quadratic on sparse networks. This fast version is competitive from a computational point of view with other hierarchical fast algorithms, and, in general, it provides better results.

翻译：本文正式将分层聚类网络问题（HCNP）定义为寻找网络优质层次划分的问题。该新问题聚焦于聚类的动态过程，而非聚类结果的最终形态。为解决此问题，我们提出了一种基于新最短路径介数度量的网络层次聚类算法。在计算该度量时，每对节点间的通信权重由建立该通信的节点重要性决定。每对节点对应的权重通过博弈的沙普利值计算，该博弈被命名为线性模块度博弈。这种新度量（节点博弈最短路径介数度量）通过消除具有最高值的边来获得网络的层次划分。为评估算法性能，我们从模块度和同质性两个视角引入多个标准，用于比较网络的不同树状图。最后，我们提出一种基于节点博弈最短路径介数度量简化版本的快速算法，该算法在稀疏网络上的复杂度为二次阶。此快速版本在计算复杂度上与其他快速层次算法具有竞争力，且通常能提供更优结果。