  We propose and study a model of scale-free growing networks that gives a
degree distribution dominated by a power-law behavior with a model-dependent,
hence tunable, exponent. The model represents a hybrid of the growing networks
based on popularity-driven and fitness-driven preferential attachments. As the
network grows, a newly added node establishes $m$ new links to existing nodes
with a probability $p$ based on popularity of the existing nodes and a
probability $1-p$ based on fitness of the existing nodes. An explicit form of
the degree distribution $P(p,k)$ is derived within a mean field approach. For
reasonably large $k$, $P(p,k) \sim k^{-\gamma(p)}{\cal F}(k,p)$, where the
function ${\cal F}$ is dominated by the behavior of $1/\ln(k/m)$ for small
values of $p$ and becomes $k$-independent as $p \to 1$, and $\gamma(p)$ is a
model-dependent exponent. The degree distribution and the exponent $\gamma(p)$
are found to be in good agreement with results obtained by extensive numerical
simulations.
