Spectral approach to finding communities in networks based on the modularity density

To detect the community structure in complex networks effectively,the modularity density function(D value) is optimized by the optimizing process,how the optimization of the D function can be reformulated as a spectral relaxation problem is proved and a new spectral clustering algorithm is proposed.The algorithm allows automatic selection of the number of community structures.The approach is illustrated and compared with the direct kernel approach based on the modularity density and spectral clustering based on modularity(Q) by using a classic computer generated networks and a real world network.Experimental results show the significance of the proposed approach,particularly,in the cases when the community structure is obscure.