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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.
A message passing algorithm is derived for recovering communities within a graph generated by a variation of the Barab{a}si-Albert preferential attachment model. The estimator is assumed to know the arrival times, or order of attachment, of the verti
We consider learning of fundamental properties of communities in large noisy networks, in the prototypical situation where the nodes or users are split into two classes according to a binary property, e.g., according to their opinions or preferences
Time-stamped data are increasingly available for many social, economic, and information systems that can be represented as networks growing with time. The World Wide Web, social contact networks, and citation networks of scientific papers and online
Community detection or clustering is a crucial task for understanding the structure of complex systems. In some networks, nodes are permitted to be linked by either positive or negative edges; such networks are called signed networks. Discovering com
A set $D$ of vertices of a graph $G$ is a total dominating set if every vertex of $G$ is adjacent to at least one vertex of $D$. The total domination number of $G$ is the minimum cardinality of any total dominating set of $G$ and is denoted by $gamma