Analysis of a Model for Generating Weakly Scale-free Networks

It is commonly believed that real networks are scale-free and the fraction of nodes P(k) with degree k satisfies the power-law P(k) proportional to k(-gamma) for k > k(min) > 0. Preferential attachment is the mechanism that has been considered responsible for such organization of these networks. In many real networks, degree distribution before the k(min) varies very slowly to the extent of being uniform as compared to the degree distribution for k > k(min). In this paper, we propose a model that describes this particular degree distribution for the whole range of k > 0. We adopt a two step approach. In the first step, at every time stamp we add a new node to the network and attach it to an existing node using preferential attachment method. In the second step, we add edges between existing pairs of nodes with the node selection based on the uniform probability distribution. Our approach generates weakly scale-free networks that closely follow the degree distribution of real-world networks. We perform a comprehensive mathematical analysis of the model in the discrete domain and compare the degree distribution generated by this model with that of real-world networks.