  We study dynamics of entanglement in the thermalization process of an
isolated quantum many-body system. We propose a simple setup for measuring the
propagation speed of entanglement entropy (EE) in numerical simulations and
apply it to the integrable/non-integrable spin models in 1D - the transverse
Ising (TI) model, the chaotic Ising (CI) model, and the extended chaotic Ising
(ECI) model. We find that two distinct time-scales $t^\ast$ and $t_{\rm
{diff}}$ arise in the dynamics of EE in the thermalization process: the former
represents the time-scale for the saturation of EE and the latter characterizes
spreading of EE over the entire system. Evaluating the propagation speed of
entanglement from $t_{\rm diff}$, we find that entanglement propagates
ballistically with a constant velocity irrespective of the integrability of the
model. The propagation speed of entanglement is found to coincide with the
maximum group velocity of quasi-particle excitations in the TI model. We also
evaluate the propagation speed of entanglement by mutual information and find
the characteristic time-scale $t_{\mathrm{MI}}$. We show that the propagation
speeds of entanglement evaluated by $t_{\mathrm{MI}}$ and $t_{\rm {diff}}$
agree well. We discuss the condition for thermalization based on the numerical
results and propose that scrambling of the entire system has to take place
before saturation of EE for thermalization.
