We show how a network of interconnections between nodes can be constructed to have a specified distribution of nodal degrees. This is achieved by treating the network as a thermodynamic system subject to constraints and then rewiring the system to maintain the constraints while increasing the entropy. The general construction is given and illustrated by the simple example of an exponential network. By considering the constraints as a cost function analogous to an internal energy, we obtain a characterisation of the correspondence between the intensive and extensive variables of the network. Applied to networks in living organisms, this approach may lead to macroscopic variables useful in characterising living systems.