  We introduce the concept of community trees that summarizes topological
structures within a network. A community tree is a tree structure representing
clique communities from the clique percolation method (CPM). The community tree
also generates a persistent diagram. Community trees and persistent diagrams
reveal topological structures of the underlying networks and can be used as
visualization tools. We study the stability of community trees and derive a
quantity called the total star number (TSN) that presents an upper bound on the
change of community trees. Our findings provide a topological interpretation
for the stability of communities generated by the CPM.
