  We introduce Tsallis mapping in Bianconi-Barab\'asi (B-B) fitness model of
growing networks. This mapping addresses the dynamical behavior of the fitness
model within the framework of nonextensive statistics mechanics, which is
characterized by a dimensionless nonextensivity parameter $q$. It is found that
this new phenomenological parameter plays an important role in the evolution of
networks: the underlying evolving networks may undergo a different phases
depending on the $q$ exponents, comparing to the original B-B fitness model,
and the corresponding critical transition "temperature" is identified.
