Using finite-size scaling, we have investigated the percolation phase transitions of evolving random networks under a generalized Achlioptas process (GAP). During this GAP, the edge with a minimum product of two connecting cluster sizes is taken with a probability p from two randomly chosen edges. This model becomes the Erdös-Rényi network at p=0.5 and the random network under the Achlioptas process at p=1. Using both the fixed point of the size ratio s 2/s 1 and the straight line of lns 1, where s 1 and s 2 are the reduced sizes of the largest and the second-largest cluster, we demonstrate that the phase transitions of this model are continuous for 0.5≤p≤1. From the slopes of lns 1 and ln (s 2 /s 1 ) ′ at the critical point, we get critical exponents β and ν of the phase transitions. At 0.5≤p≤0.8, it is found that β, ν, and s 2/s 1 at critical point are unchanged and the phase transitions belong to the same universality class. When p≥0.9, β, ν, and s 2/s 1 at critical point vary with p and the universality class of phase transitions depends on p.