  We demonstrate that with appropriate quantum correlation function, a
real-space network model can be constructed to study the phase transitions in
quantum systems. For the three-dimensional bosonic system, the single-particle
density matrix is adopted to construct the adjacency matrix. We show that the
Bose-Einstein condensate transition can be interpreted as the transition into a
small-world network, which is accurately captured by the small-world
coefficient. For the one-dimensional disordered system, using the electron
diffusion operator to build the adjacency matrix, we find that the Anderson
localized states create many weakly-linked subgraphs, which significantly
reduces the clustering coefficient and lengthens the shortest path. We show
that the crossover from delocalized to localized regimes as a function of the
disorder strength can be identified as the loss of global connection, which is
revealed by the small-world coefficient as well as other independent measures
like the robustness, the efficiency, and the algebraic connectivity. Our
results suggest that the quantum phase transitions can be visualized in real
space and characterized by the network analysis with suitable choices of
quantum correlation functions.
