  We extend the Bianconi-Barab\'asi (B-B) fitness model to the non-growing
complex network with fixed number of nodes and links. It is found that the
statistical physics of this model makes it an appropriate representation of the
Boltzmann statistics in the context of complex networks. The phase transition
of this extended model is illustrated with numerical simulation and the
corresponding "critical temperature" $T_c$ is identified. We note that the
"non-condensation phase" in $T > T_c$ regime is different with "fit-get-rich"
(FGR) phase of B-B model and that the connectivity degree distribution P(k)
deviates from power-law distribution at given temperatures.
