  We study complex networks formed by triangulations and higher-dimensional
simplicial complexes representing closed evolving manifolds. In particular, for
triangulations, the set of possible transformations of these networks is
restricted by the condition that at each step, all the faces must be triangles.
Stochastic application of these operations leads to random networks with
different architectures. We perform extensive numerical simulations and explore
the geometries of growing and equilibrium complex networks generated by these
transformations and their local structural properties. This characterization
includes the Hausdorff and spectral dimensions of the resulting networks, their
degree distributions, and various structural correlations. Our results reveal a
rich zoo of architectures and geometries of these networks, some of which
appear to be small worlds while others are finite-dimensional with Hausdorff
dimension equal or higher than the original dimensionality of their simplices.
The range of spectral dimensions of the evolving triangulations turns out to be
from about 1.4 to infinity. Our models include simplicial complexes
representing manifolds with evolving topologies, for example, an h-holed torus
with a progressively growing number of holes. This evolving graph demonstrates
features of a small-world network and has a particularly heavy-tailed degree
distribution.
