  Many networks generated by nature have two generic properties: they are
formed in the process of {preferential attachment} and they are scale-free.
Considering these features, by interfering with mechanism of the {preferential
attachment}, we propose a generalisation of the Barab\'asi--Albert model---the
'Fractional Preferential Attachment' (FPA) scale-free network model---that
generates networks with time-independent degree distributions $p(k)\sim
k^{-\gamma}$ with degree exponent $2<\gamma\leq3$ (where $\gamma=3$ corresponds
to the typical value of the BA model). In the FPA model, the element
controlling the network properties is the $f$ parameter, where $f \in
(0,1\rangle$. Depending on the different values of $f$ parameter, we study the
statistical properties of the numerically generated networks. We investigate
the topological properties of FPA networks such as degree distribution, degree
correlation (network assortativity), clustering coefficient, average node
degree, network diameter, average shortest path length and features of
fractality. We compare the obtained values with the results for various
synthetic and real-world networks. It is found that, depending on $f$, the FPA
model generates networks with parameters similar to the real-world networks.
Furthermore, it is shown that $f$ parameter has a significant impact on, among
others, degree distribution and degree correlation of generated networks.
Therefore, the FPA scale-free network model can be an interesting alternative
to existing network models. In addition, it turns out that, regardless of the
value of $f$, FPA networks are not fractal.
