  With the aim of considering models with persistent memory we propose a
fractional nonlinear modification of the classical Yule model often studied in
the context of macrovolution. Here the model is analyzed and interpreted in the
framework of the development of networks such as the World Wide Web.
Nonlinearity is introduced by replacing the linear birth process governing the
growth of the in-links of each specific webpage with a fractional nonlinear
birth process with completely general birth rates.
  Among the main results we derive the explicit distribution of the number of
in-links of a webpage chosen uniformly at random recognizing the contribution
to the asymptotics and the finite time correction. The mean value of the latter
distribution is also calculated explicitly in the most general case.
Furthermore, in order to show the usefulness of our results, we particularize
them in the case of specific birth rates giving rise to a saturating behaviour,
a property that is often observed in nature. The further specialization to the
non-fractional case allows us to extend the Yule model accounting for a
nonlinear growth.
