  The maximum capacity of fractal D2D (device-to-device) social networks with
both direct and hierarchical communications is studied in this paper.
Specifically, the fractal networks are characterized by the direct social
connection and the self-similarity. Firstly, for a fractal D2D social network
with direct social communications, it is proved that the maximum capacity is $
\Theta\left(\frac{1}{\sqrt{n\log n}}\right) $ if a user communicates with one
of his/her direct contacts randomly, where $ n $ denotes the total number of
users in the network, and it can reach up to $ \Theta\left(\frac{1}{\log
n}\right) $ if any pair of social contacts with distance $ d $ communicate
according to the probability in proportion to $ d^{-\beta} $. Secondly, since
users might get in touch with others without direct social connections through
the inter-connected multiple users, the fractal D2D social network with these
hierarchical communications is studied as well, and the related capacity is
further derived. Our results show that this capacity is mainly affected by the
correlation exponent $\epsilon$ of the fractal structure. The capacity is
reduced in proportional to $ \frac{1}{{\log n}} $ if $ 2<\epsilon<3 $, while
the reduction coefficient is $ \frac{1}{n} $ if $ \epsilon=3 $.
