Generation of scale-free networks using a simple preferential-rewiring dynamics

We propose a simple dynamical process for non-growing networks, where steady states in the long-time limit exhibit power-law degree distributions with the exponent 2. At each time step, two nodes, ii and jj, are randomly selected, and one incoming link to ii is redirected to jj with the rewiring probability R  , determined only by degrees of two nodes, kiki and kjkj, while higher-degree nodes are preferred to get another link. This is an application of the general model introduced earlier [S. Ree, Phys. Rev. E 73 (2006) 026115]. To take the structure of networks into account, we also consider three possible distinctions for the model: (i) how we choose a rewiring link out of all incoming links to ii (three cases), (ii) whether links are directed or not (two cases), (iii) types of networks considering the existence of self-loops and multiple links (two cases); as a result, we specify the total of 12 different cases of the model. We then observe numerically that networks will evolve to steady states with power-law degree distributions when parameters of the model satisfy certain conditions. This work is from an effort to find a simple model of the network dynamics generating scale-free networks, and has a potential to become an underlying mechanism for wide range of scale-free non-growing networks.