  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.
