  In this paper, we propose a method that extends the persistence-based
topological data analysis (TDA) that is typically used for characterizing
shapes to general networks. We introduce the concept of the community tree, a
tree structure established based on clique communities from the clique
percolation method, to summarize the topological structures in a network from a
persistence perspective. Furthermore, we develop efficient algorithms to
construct and update community trees by maintaining a series of clique graphs
in the form of spanning forests, in which each spanning tree is built on an
underlying Euler Tour tree. With the information revealed by community trees
and the corresponding persistence diagrams, our proposed approach is able to
detect clique communities and keep track of the major structural changes during
their evolution given a stability threshold. The results demonstrate its
effectiveness in extracting useful structural insights for time-varying social
networks.
