A Note on Community Trees in Networks

TL;DR

This paper introduces community trees as a topological visualization tool for network communities, derived from clique percolation, and analyzes their stability using persistent diagrams and the total star number.

Contribution

It presents a novel topological framework for community detection in networks, including the concept of community trees and their stability analysis.

Findings

Community trees effectively visualize clique communities.

The stability of community trees is bounded by the total star number.

Persistent diagrams reveal topological structures of networks.

Abstract

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.