  In this paper, we present a rigorous derivation of the mean-field limit for a
moderately interacting particle system in $\R^d$ $(d\geq 2)$. For stochastic
initial data, we demonstrate that the solution to the interacting particle
model, with an appropriately applied cut-off, converges in probabilistic sense
to the solution of the characteristics of the regularized Vlasov models
featuring local alignments and Newtonian potential. Notably, the cutoff
parameter for the singular potential is selected to scale polynomially with the
number of particles, representing an improvement over the logarithmic cut-off
obtained in [38].
