  We consider single-particle quantum transport on parametrized complex
networks. Based on general arguments regarding the spectrum of the
corresponding Hamiltonian, we derive bounds for a measure of the global
transport efficiency defined by the time-averaged return probability. For
tree-like networks, we show analytically that a transition from efficient to
inefficient transport occurs depending on the (average) functionality of the
nodes of the network. In the infinite system size limit, this transition can be
characterized by an exponent which is universal for all tree-like networks. Our
findings are corroborated by analytic results for specific deterministic
networks, dendrimers and Viscek fractals, and by Monte Carlo simulations of
iteratively built scale-free trees.
