Quasistatic scale-free networks

A network is formed using the N sites of a one-dimensional lattice in the shape of a ring as nodes and each node with the initial degree $k_{\mathrm{in}}=2.\mathrm{}$ N links are then introduced 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 $〈k_m(\alpha ,N)〉$ changes its variation from $N^0$ to N at a specific transition point of ${\alpha }_c.$ The network is scale free, i.e., the nodal degree distribution has a power law decay for $\alpha >~{\alpha }_c.$