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Revealing Cluster Structure of Graph by Path Following Replicator Dynamic

In this paper, we propose a path following replicator dynamic, and investigate its potentials in uncovering the underlying cluster structure of a graph. The proposed dynamic is a generalization of the discrete replicator dynamic. The replicator dynamic has been successfully used to extract dense clusters of graphs; however, it is often sensitive to the degree distribution of a graph, and usually biased by vertices with large degrees, thus may fail to detect the densest cluster. To overcome this problem, we introduce a dynamic parameter, called path parameter, into the evolution process. The path parameter can be interpreted as the maximal possible probability of a current cluster containing a vertex, and it monotonically increases as evolution process proceeds. By limiting the maximal probability, the phenomenon of some vertices dominating the early stage of evolution process is suppressed, thus making evolution process more robust. To solve the optimization problem with a fixed path parameter, we propose an efficient fixed point algorithm. The time complexity of the path following replicator dynamic is only linear in the number of edges of a graph, thus it can analyze graphs with millions of vertices and tens of millions of edges on a common PC in a few minutes. Besides, it can be naturally generalized to hypergraph and graph with edges of different orders. We apply it to four important problems: maximum clique problem, densest k-subgraph problem, structure fitting, and discovery of high-density regions. The extensive experimental results clearly demonstrate its advantages, in terms of robustness, scalability and flexility.