On the Capacity of Fractal Wireless Networks With Direct Social Interactions

Abstract: The capacity of a fractal wireless network with direct social interactions 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 connection feature between two nodes with degree $ k_{1} $ and $ k_{2} $ by a joint probability distribution $ P(k_{1},k_{2}) $. It is proved that if the source node communicates with one of its direct contacts randomly, the maximum capacity is consistent with the classical result $ \Theta\left(\frac{1}{\sqrt{n\log n}}\right) $ achieved by Kumar \cite{Gupta2000The}. On the other hand, if the two nodes with distance $ d $ communicate according to the probability $ d^{-\beta} $, the maximum capacity can reach up to $ \Theta\left(\frac{1}{\log n}\right) $, which exhibits remarkable improvement compared with the well-known result in \cite{Gupta2000The}.