The similarity of weights on edges and discovering of community structure

In this paper, we propose a weighted modularity QW based on the similarity of weights on edges and a threshold coefficient ζ to evaluate the equivalence of edge weights. Simulations on benchmark networks and real networks show that optimization on the modularity enable us to obtain groups of nodes within which the edge weights are distributed uniformly but at random between them. The communities can reveal the uniform connections (stable relationships measured by the similarity of weights on edges) between nodes or some similarity between nodes' functions. Furthermore, with the dynamical moving of ζ, we observe that optimization on the QW allows for the discovering of a special hierarchical organization which reveals different levels of uniform connections between nodes in networks. The substructures revealed by the hierarchical organization enable us to obtain more information of networks, and give a potential way for partly remedying the intrinsic resolution problem of modularity.