Explore of the fuzzy community structure integrating the directed line graph and likelihood optimization

Abstract

Detecting fuzzy network communities in directed network is a classic and very difficult task in the field of complex network analysis, principally for its applications in domains such as social or biological networks analysis. Present techniques rely heavily on network topology, which cannot provide a lot of important information, such as module correlation and hierarchical structure. In this paper, we present a new fuzzy community detection method, which is able to find fuzzy communities in directed line graphs by maximizing likelihood function. Firstly, the directed node graph is transformed to a new type directed line graph, and the direction and weight of line graph are defined. Then, the community unit consists of membership and correlation information is defined in the line graph. Specifically, there are two main contributions of this method: 1) to adequately characterize the community structure, the node and module correlation with different granularity can be calculated; 2) based on the membership and correlation information, we can extract the multiplex patterns between communities, according to different domain requirements. Furthermore, we are able to map the link community configuration to the optimal situation dynamically by maximizing the likelihood function with rigorous mathematical proof. Based on the spectral analysis of the Markovian transition matrix, a mathematical theory is provided to identify the optimum number of network communities, and to analyze the stability of the community structure. Extensive simulations using both synthetic and real-world benchmark networks are performed to verify the algorithmic performance.