Degree distributions in optimized directed network models: Emergence of scale-free networks

We consider the recently proposed growing directed network models that aim at minimizing the weighted connection expenses while at the same time favoring weighted local inward and outward node degrees. Theoretical expressions for the in-degree and out-degree distributions of the optimized networks are derived for general edge weight as well as inward and outward node weight distributions. In particular, it is shown that for exponentially distributed node weights and narrow edge weight distributions, the emerged optimized degree distributions resulted in scale-free power-law decay in broad regimes. Furthermore, it is shown that the exponents of the scale-free distribution are related to the mean values of the node weight distributions in a universal way. Simulations explicitly verify all the theoretical results. Implications of the results are also discussed and compared with data in realistic networks, suggesting that the ratio of the out-degree power-law exponent to that of the in-degree can possibly reveal some intrinsic characteristic features of the node population.