  Aims. We investigate how properties of the ensemble of superclusters in the
cosmic web evolve with time. Methods. We perform numerical simulations of the
evolution of the cosmic web using the LambdaCDM model in box sizes L0 = 1024,
512, 256 Mpc/h. We find supercluster ensembles of models for four evolutionary
stages, corresponding to the present epoch z = 0, and to redshifts z = 1, z =
3, and z = 10. We calculate fitness diameters of superclusters defined from
volumes of superclusters divided to filling factors of over-density regions.
Geometrical and fitness diameters of largest superclusters, and the number of
superclusters as functions of the threshold density are used as percolation
functions to describe geometrical properties of the ensemble of superclusters
in the cosmic web. We calculate distributions of geometrical and fitness
diameters and luminosities of superclusters, and follow time evolution of
percolation functions and supercluster distributions. We compare percolation
functions and supercluster distributions of models and samples of galaxies of
the Sloan Digital Sky Survey (SDSS). Results. Our analysis shows that fitness
diameters of superclusters have a minimum at certain threshold density. Fitness
diameters around minima almost do not change with time in co-moving
coordinates. Numbers of superclusters have maxima which are approximately
constant for all evolutionary epochs. Geometrical diameters of superclusters
decrease during the evolution of the cosmic web; luminosities of superclusters
increase during the evolution. Conclusions. Our study suggests that
evolutionary changes occur inside dynamical volumes of superclusters. The
stability of fitness diameters and numbers of superclusters during the
evolution is an important property of the cosmic web.
