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, which evolves to a local maximum of the optimization problem max f(x) = xWx,x ∈ ∆ (W is a non-negative square matrix and ∆ is a simplex), and has been widely used to stimulate the evolution process of animal behavior. The discrete 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 ε, called path parameter, into the evolution process. That is, we successively solve a series of optimization problems, max f(x) = xWx,x ∈ ∆ε, where ∆ε = {x| ∑ i xi = 1, xi ∈ [0, ε]}. The path parameter ε 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 ε, we propose an efficient fixed point algorithm. Intuitively, the proposed dynamic follows the solution path of the optimization problems max f(x) = xWx,x ∈ ∆ε, with gradually expanding domain ∆ε. The key properties of the proposed path following replicator dynamic are: 1) its probability to evolve to the most significant cluster of graph is much higher than discrete replicator dynamic, 2) the path parameter ε offers us a tool to control the evolution process, and we can use it to simultaneously obtain dense subgraphs of various specified sizes, and 3) the evolution process is essentially a shrink process of high-density regions, thus reveals the underlying cluster structure of graph. 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, where f(x) becomes a polynomial function. 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.