  Many stochastic complex systems are characterized by the fact that their
configuration space doesn't grow exponentially as a function of the degrees of
freedom. The use of scaling expansions is a natural way to measure the
asymptotic growth of the configuration space volume in terms of the scaling
exponents of the system. These scaling exponents can, in turn, be used to
define universality classes that uniquely determine the statistics of a system.
Every system belongs to one of these classes. Here we derive the information
geometry of scaling expansions of sample spaces. In particular, we present the
deformed logarithms and the metric in a systematic and coherent way. We observe
a phase transition for the curvature. The phase transition can be well measured
by the characteristic length r, corresponding to a ball with radius 2r having
the same curvature as the statistical manifold. Increasing characteristic
length with respect to the size of the system is associated with
sub-exponential sample space growth is associated with strongly constrained and
correlated complex systems. Decreasing of the characteristic length corresponds
to super-exponential sample space growth that occurs for example in systems
that develop structure as they evolve. Constant curvature means exponential
sample space growth that is associated with multinomial statistics, and
traditional Boltzmann-Gibbs, or Shannon statistics applies. This allows us to
characterize transitions between statistical manifolds corresponding to
different families of probability distributions.
