An index of network evolution speed and a network evolution model were put forward to analyze the effects of network evolution speed on propagation. The definition of temporal correlation coefficient was modified to characterize the speed of the network evolution; meanwhile, a non-Markov model of temporal networks was proposed. For every active node at a time step, a random node from network was selected with probability r, while a random node from former neighbors of the active node was selected with probability 1-r. Edges were created between the active node and its corresponding selected nodes. The simulation results confirm that there is a monotone increasing relationship between the network model parameter r and the network evolution speed; meanwhile, the greater the value of r, the greater the scope of the spread on network becomes. These mean that the temporal networks with high evolution speed are conducive to the spread on networks. More specifically, the rapidly changing network topology is conducive to the rapid spread of information, but not conducive to the suppression of virus propagation.