  We examine the fundamental question whether a random discrete structure with
the minimal number of restrictions can converge to continuous metric space. We
study the geometrical properties such as the dimensionality and the curvature
emerging out of the connectivity properties of uniform random graphs. In
addition we introduce a simple evolution mechanism for the graph by removing
one edge per a fundamental quantum of time from an initially complete graph. We
show an exponential growth of the radius of the graph, that ends up in a random
structure with emergent average spatial dimension $D=3$ and zero curvature
$K=0$, resembling a flat 3D manifold, that could describe the observed space in
our universe and some of its geometrical properties. In addition, we introduce
a generalized action for graphs based on physical quantities on different
subgraph structures that helps to recover the well known properties of
spacetime as described in general relativity, like time dilation due to
gravity. Also, we show how various quantum mechanical concepts such as
generalized uncertainty principles based on the statistical fluctuations can
emerge from random discrete models. Moreover, our approach leads to a
unification of space and matter-energy, for which we propose a
mass-energy-space equivalence that leads to a way to transform between empty
space and matter-energy via the cosmological constant.
