  Critical phenomena can show unusual phase diagrams when defined in complex
network topologies. The case of classical phase transitions such as the
classical Ising model and the percolation transition has been studied
extensively in the last decade. Here we show that the phase diagram of the
Bose-Hubbard model, an exclusively quantum mechanical phase transition, also
changes significantly when defined on random scale-free networks. We present a
mean-field calculation of the model in annealed networks and we show that when
the second moment of the average degree diverges the Mott-insulator phase
disappears in the thermodynamic limit. Moreover we study the model on quenched
networks and we show that the Mott-insulator phase disappears in the
thermodynamic limit as long as the maximal eigenvalue of the adjacency matrix
diverges. Finally we study the phase diagram of the model on Apollonian
scale-free networks that can be embedded in 2 dimensions showing the extension
of the results also to this case.
