  Inspired by "quantum graphity" models for spacetime, a statistical model of
graphs is proposed to explore possible realizations of emergent manifolds.
Graphs with given numbers of vertices and edges are considered, governed by a
very general Hamiltonian that merely favors graphs with near-constant valency
and local rotational symmetry. The ratio of vertices to edges controls the
dimensionality of the emergent manifold. The model is simulated numerically in
the canonical ensemble for a given vertex to edge ratio, where it is found that
the low-energy states are almost triangulations of two-dimensional manifolds.
The resulting manifold shows topological "handles" and surface intersections in
a higher embedding space, as well as non-trivial fractal dimension consistent
with previous spectral analysis, and nonlocal links consistent with models of
disordered locality. The transition to an emergent manifold is first order, and
thus dependent on microscopic structure. Issues involved in interpreting
nearly-fixed valency graphs as Feynman diagrams dual to a triangulated manifold
as in matrix models are discussed. Another interesting phenomenon is that the
entropy of the graphs are super-extensive, a fact known since Erd\H{o}s, which
results in a transition temperature of zero in the limit of infinite system
size: infinite manifolds are always disordered. Aside from a finite universe or
diverging coupling constraints as possible solutions to this problem,
long-range interactions between vertex defects also resolve the problem and
restore a nonzero transition temperature, in a manner similar to that in
low-dimensional condensed-matter systems.
