This paper aims to establish explicit relationships between the controllability degree of a network, that is, the control energy required to move the network between different states, and its graphical structure and edge weights. As it is extremely challenging to accomplish this task for general networks, we focus on the case where the network controllability Gramian is a diagonal matrix. The main technical contributions of the paper are (i) to derive necessary and sufficient graphical conditions for networks to feature a diagonal controllability Gramian, and (ii) to propose a constructive algorithm to design network topologies and weights so as to generate stable and controllable networks with pre-specified diagonal Gramians. The proposed network design algorithm allows for individual assignment of how each node responds to external stimuli, so as to selectively enforce robustness to external disturbances. While relying on the simplifying assumption of a diagonal controllability Gramian, our analysis reveals novel and counterintuitive controllability properties of complex networks. For instance, we identify a class of continuous-time networks where the control energy is independent of their cardinality and number of control nodes (thus disproving existing results based on numerical controllability studies), discuss their stability margin, and show that the energy required to control a node can be made independent of its graphical distance from the control nodes. These results complement and formally support, or challenge, a series of conjectures based on numerical studies in the field of complex networks. (C) 2019 Elsevier Ltd. All rights reserved.