  In this paper we focus on the problem of the degree sequence for the
following random graph process. At any time-step $t$, one of the following
three substeps is executed: with probability $\alpha_1$, a new vertex $x_t$ and
$m$ edges incident with $x_t$ are added; or, with probability
$\alpha-\alpha_1$, $m$ edges are added; or finally, with probability $1-\a$,
$m$ random edges are deleted. Note that in any case edges are added in the
manner of preferential attachment. we prove that there exists a critical point
$\alpha_c$ satisfying: 1) if $\alpha_1<\alpha_c$, then the model has power law
degree sequence; 2) if $\alpha_1>\alpha_c$, then the model has exponential
degree sequence; and 3) if $\alpha_1=\alpha_c$, then the model has a degree
sequence lying between the above two cases.
