Uniform Preferential Selection Model for Generating Scale-free Networks

It has been observed in real networks that the fraction of nodes P(k) with degree k satisfies the power-law P(k) proportional to k(-gamma) for k > k(min) > 0. However, the degree distribution of nodes in these networks before k(min) varies slowly to the extent of being uniform as compared to the degree distribution after k(min). Most of the previous studies focus on the degree distribution after k(min) and ignore the initial flatness in the distribution of degrees. In this paper, we propose a model that describes the degree distribution for the whole range of k > 0, i.e., before and after k(min). The network evolution is made up of two steps. In the first step, a new node is connected to the network through a preferential attachment method. In the second step, a certain number of edges between the existing nodes are added such that the end nodes of an edge are selected either uniformly or preferentially. The model has a parameter to control the uniform or preferential selection of nodes for creating edges in the network. We perform a comprehensive mathematical analysis of our proposed model in the discrete domain and prove that the model exhibits an asymptotically power-law degree distribution after k(min) and a flat-ish distribution before k(min). We also develop an algorithm that guides us in determining the model parameters in order to fit the model output to the node degree distribution of a given real network. Our simulation results show that the degree distributions of the graphs generated by this model match well with those of the real-world graphs.