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) ∝ k^−γ 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. [ABSTRACT FROM AUTHOR]