  We investigate the evolution of superclusters and supercluster cocoons
(basins of attraction), and the influence of cosmological parameters to the
evolution. We perform numerical simulations of the evolution of the cosmic web
for different cosmological models: the LCDM model with a conventional value of
the dark energy (DE) density, the open model OCDM with no DE, the standard SCDM
model with no DE, and the Hyper-DE HCDM model with an enhanced DE density
value. We find ensembles of superclusters of these models for five evolutionary
stages, corresponding to the present epoch z = 0, and to redshifts z = 1, 3,
10, 30. We use diameters of the largest superclusters and the number of
superclusters as percolation functions to describe properties of the ensemble
of superclusters in the cosmic web. We analyse the size and mass distribution
of superclusters in models and in real Sloan Digital Sky Survey (SDSS) based
samples. In all models numbers and volumes of supercluster cocoons are
independent on cosmological epochs. Supercluster masses increase with time, and
geometrical sizes in comoving coordinates decrease with time, for all models.
LCDM, OCDM and HCDM models have almost similar percolation parameters. This
suggests that the essential parameter, which defines the evolution of
superclusters, is the matter density. The DE density influences the growth of
the amplitude of density perturbations, and the growth of masses of
superclusters, albeit significantly less strongly. The HCDM model has the
largest speed of the growth of the amplitude of density fluctuations, and the
largest growth of supercluster masses during the evolution. Geometrical
diameters and numbers of HCDM superclusters at high threshold densities are
larger than for LCDM and OCDM superclusters. SCDM model has about two times
more superclusters than other models; SCDM superclusters have smaller diameters
and masses.
