  We investigate the basic properties of voids from high resolution,
cosmological N-body simulations of {\Lambda}-dominated cold dark matter
({\Lambda}CDM) models, in order to compare with the analytical model of Sheth
and van de Weygaert (SvdW) for void statistics. For the subsample of five dark
matter simulations in the {\Lambda}CDM cosmology with box sizes ranging from
1000 Mpc/h to 8 Mpc/h, we find that the standard void-in-cloud effect is too
simplified to explain several properties of identified small voids in
simulations. (i) The number density of voids is found to be larger than the
prediction of the analytical model up to 2 orders of magnitude below 1 Mpc/h
scales. The Press-Schechter model with the linear critical threshold of void
{\delta}_v = -2.71, or a naive power law, is found to provide an excellent
agreement with the void size function, suggesting that the void-in-cloud effect
does not suppress as much voids as predicted by the SvdW model. (ii) We then
measured the density and velocity profiles of small voids, and find that they
are mostly partially collapsing underdensities, instead of being completely
crushed in the standard void-in-cloud scenario. (iii) Finally, we measure the
void distributions in four different tidal environments, and find that the
void-in-void effect alone can explain the correlation between distribution and
environments, whereas the void-in-cloud effect is only weakly influencing the
abundance of voids, even in filaments and clusters.
