  We generalize an algorithm used widely in the configuration model such that
power-law degree sequences with the degree exponent $\lambda$ and the number of
links per node $K$ controllable independently may be generated. It yields the
degree distribution in a different form from that of the static model or under
random removal of links while sharing the same $\lambda$ and $K$. With this
generalized power-law degree distribution, the critical point $K_c$ for the
appearance of the giant component remains zero not only for $\lambda\leq 3$ but
also for $3<\lambda<\lambda_l \simeq 3.81$. This is contrasted with $K_c=0$
only for $\lambda\leq 3$ in the static model and under random link removal. The
critical exponents and the cluster-size distribution for $\lambda<\lambda_l$
are also different from known results. By analyzing the moments and the
generating function of the degree distribution and comparison with those of
other models, we show that the asymptotic behavior and the degree exponent may
not be the only properties of the degree distribution relevant to the critical
phenomena but that its whole functional form can be relevant. These results can
be useful in designing and assessing the structure and robustness of networked
systems.
