  In various theories of quantum gravity, one observes a change in the spectral
dimension from the topological spatial dimension $d$ at large length scales to
some smaller value at small, Planckian scales. While the origin of such a flow
is well understood in continuum approaches, in theories built on discrete
structures a firm control of the underlying mechanism is still missing. We shed
some light on the issue by presenting a particular class of quantum geometries
with a flow in the spectral dimension, given by superpositions of states
defined on regular complexes. For particular superposition coefficients
parametrized by a real number $0<\alpha<d$, we find that the spatial spectral
dimension reduces to $d_s \simeq \alpha$ at small scales. The spatial Hausdorff
dimension of such class of states varies between 1 and $d$, while the walk
dimension takes the usual value $d_w=2$. Therefore, these quantum geometries
may be considered as fractal only when $\alpha=1$, where the "magic number"
${d_s}^{\rm spacetime}\simeq 2$ for the spectral dimension of space\emph{time},
appearing so often in quantum gravity, is reproduced as well. These results
apply, in particular, to special superpositions of spin-network states in loop
quantum gravity, and they provide more solid indications of dimensional flow in
this approach.
