  New entropy measures have been recently introduced for the quantification of
the complexity of networks. Most of these entropy measures apply to static
networks or to dynamical processes defined on static complex networks. In this
paper we define the entropy rate of growing network models. This entropy rate
quantifies how many labeled networks are typically generated by the growing
network models. We analytically evaluate the difference between the entropy
rate of growing tree network models and the entropy of tree networks that have
the same asymptotic degree distribution. We find that the growing networks with
linear preferential attachment generated by dynamical models are exponentially
less than the static networks with the same degree distribution for a large
variety of relevant growing network models. We study the entropy rate for
growing network models showing structural phase transitions including models
with non-linear preferential attachment. Finally, we bring numerical evidence
that the entropy rate above and below the structural phase transitions follow a
different scaling with the network size.
