Static Scale-free Networks

A network is formed using the N sites of an one-dimensional lattice in the shape of a ring as nodes and with the initial attractiveness k_o = 2, the degree of each node. N links are then added to this network, each link starts from a distinct node, the other end being connected to any other node with degree k randomly selected with an attachment probability proportional to k^{alpha}. Tuning the control parameter alpha we observe a transition where the average degree of the largest node changes its variation from $N^0$ to N at a specific transition point of alpha_c. The degree distribution is found to be a power law only at alpha_c indicating that the network is scale-free.