Hierarchical and mixing properties of static complex networks emerging from the fluctuating classical random graphs

The Erdos-Renyi classical random graph is characterized by a fixed linking probability for all pairs of vertices. Here, this concept is generalized by drawing the linking probability from a certain distribution. Such a procedure is found to lead to a static complex network with an arbitrary connectivity distribution. In particular, a scale-free network with the hierarchical organization is constructed without assuming any knowledge about the global linking structure, in contrast to the preferential attachment rule for a growing network. The hierarchical and mixing properties of the static scale-free network thus constructed are studied. The present approach establishes a bridge between a scalar characterization of individual vertices and topology of an emerging complex network. The result may offer a clue for understanding the origin of a few abundance of connectivity distributions in a wide variety of static real-world networks.