A Note on Community Trees in Networks
This work provides a topological interpretation for community stability in networks, which is incremental as it builds on the existing clique percolation method.
The paper tackles the problem of summarizing topological structures in networks by introducing community trees, which are tree structures derived from clique communities using the clique percolation method, and it results in deriving the total star number (TSN) as an upper bound on the change of community trees to interpret stability.
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.