  Complex networks in different areas exhibit degree distributions with heavy
upper tail. A preferential attachment mechanism in a growth process produces a
graph with this feature. We herein investigate a variant of the simple
preferential attachment model, whose modifications are interesting for two main
reasons: to analyze more realistic models and to study the robustness of the
scale free behavior of the degree distribution. We introduce and study a model
which takes into account two different attachment rules: a preferential
attachment mechanism (with probability 1-p) that stresses the rich get richer
system, and a uniform choice (with probability p) for the most recent nodes.
The latter highlights a trend to select one of the last added nodes when no
information is available. The recent nodes can be either a given fixed number
or a proportion (\alpha n) of the total number of existing nodes. In the first
case, we prove that this model exhibits an asymptotically power-law degree
distribution. The same result is then illustrated through simulations in the
second case. When the window of recent nodes has constant size, we herein prove
that the presence of the uniform rule delays the starting time from which the
asymptotic regime starts to hold. The mean number of nodes of degree k and the
asymptotic degree distribution are also determined analytically. Finally, a
sensitivity analysis on the parameters of the model is performed.
