  We investigate the self-organization of point-particles with short-range
interactions modeled via simple 1D and 2D Hubbard-like models. We show how
various properties emerge such as, boson-like ordering leading to topological
structures in real space, via the clustering of the particles at discrete
energy levels, which can be analyzed using a network/graph mathematical
language. We calculate analytically the number of clusters, the correlations
between them and topological numbers like the Euler characteristic, deriving
different organizational schemes and entanglement entropy scaling laws. All
calculations are performed for an arbitrary number of particles N and energy of
the system E. Our results demonstrate how orders with diverse
topological/geometric and entanglement features, emerge in strongly interacting
many-body systems, that follow minimal microscopic interaction rules.
