  In this work, we study the evolution of Betti curves obtained by
persistent-homological analysis of pointclouds formed by halos in different
cosmological $N$-body simulations. We show that they can be approximated with a
scaled log-normal distribution function with reasonable precision. Our analysis
shows that the shapes and maximums of Betti curves exhibit dependence on the
mass range of the selected subpopulation of halos, but at the same time, the
resolution of a simulation does not play any significant role, provided that
the mass distribution of simulated halos is complete down to a given mass
scale. Besides, we study how Betti curves change with the evolution of the
Universe, i.e., their dependence on redshift. Sampling subpopulations of halos
within certain mass ranges up to redshift $z=2.5$ yields a surprisingly small
difference between corresponding Betti curves. We propose that this may be an
indicator of the existence of a new specific topological invariant in the
structure of the Universe.
