Abstract :
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, i and j, are randomly selected, and one incoming link to i is redirected to j with the rewiring probability R, determined only by degrees of two nodes, ki and kj, 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 i (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.
