Classical statistical physics has developed a powerful set of tools for analyzing complex systems, chief among them complex networks, in which connectivity and topology predominate over other system features. Complex networks model systems as diverse as the brain and the internet; however, they have had little application to quantum systems, till now. Framing connectivity in terms of quantum non-locality, we bring complex network theory into the quantum realm via an adjacency matrix constructed from mutual information, or long-range entanglement. Using matrix-product-state computational methods, we apply this new set of quantum tools to an emergent feature, quantum phase transitions. We demonstrate rapid finite size-scaling for both transverse Ising and Bose-Hubbard models, including \(Z_2\), mean field, and BKT transitions. This work opens the door for a new set of tools for complex quantum systems, as well as providing an operator-independent alternative approach to the study of critical phenomena.