On the Capacity of Wireless Networks with Fractal and Hierarchical Social Communications

The capacity of a wireless network with fractal and hierarchical social communications is studied in this paper. Specifically, we mathematically formulate the self-similarity of a fractal wireless network by a power-law degree distribution P (k), and we capture the direct social connection feature between two nodes with degree k1 and k2 by a joint probability distribution P (k1, k2). Firstly, for a fractal wireless network with direct social communications, it is proved that the maximum capacity is Θ ( 1 √ n logn ) with n denotes the total number of nodes in the network, if the source node communicates with one of its direct contacts randomly, and it can reach up to Θ ( 1 logn ) if the two nodes with distance d communicate according to the probability in proportion to d−β . Secondly, since humans might get in touch with others without direct connection but through the interconneced users, the fractal wireless networks with hierarchical social communications is studied as well, and the related capacity is derived based on the results in the case with direct social communications. Our results show that this capacity is mainly affected by the correlation exponent of the fractal networks. The capacity is reduced in proportional to 1 logn if 2 < < 3, while the reduction coefficient is 1 n if = 3.