  Recent researches on complex systems highlighted the so-called super-linear
growth phenomenon. As the system size $P$ measured as population in cities or
active users in online communities increases, the total activities $X$ measured
as GDP or number of new patents, crimes in cities generated by these people
also increases but in a faster rate. This accelerating growth phenomenon can be
well described by a super-linear power law $X \propto P^{\gamma}$($\gamma>1$).
However, the explanation on this phenomenon is still lack. In this paper, we
propose a modeling framework called growing random geometric models to explain
the super-linear relationship. A growing network is constructed on an abstract
geometric space. The new coming node can only survive if it just locates on an
appropriate place in the space where other nodes exist, then new edges are
connected with the adjacent nodes whose number is determined by the density of
existing nodes. Thus the total number of edges can grow with the number of
nodes in a faster speed exactly following the super-linear power law. The
models cannot only reproduce a lot of observed phenomena in complex networks,
e.g., scale-free degree distribution and asymptotically size-invariant
clustering coefficient, but also resemble the known patterns of cities, such as
fractal growing, area-population and diversity-population scaling relations,
etc. Strikingly, only one important parameter, the dimension of the geometric
space, can really influence the super-linear growth exponent $\gamma$.
