Detecting the fuzzy clusters of complex networks

To find the best partition of a large and complex network into a small number of clusters has been addressed in many different ways. However, the probabilistic setting in which each node has a certain probability of belonging to a certain cluster has been scarcely discussed. In this paper, a fuzzy partitioning formulation, which is extended from a deterministic framework for network partition based on the optimal prediction of a random walker Markovian dynamics, is derived to solve this problem. The algorithms are constructed to minimize the objective function under this framework. It is demonstrated by the simulation experiments that our algorithms can efficiently determine the probabilities with which a node belongs to different clusters during the learning process. Moreover, they are successfully applied to two real-world networks, including the social interactions between members of a karate club and the relationships of some books on American politics bought from Amazon.com.