  In this work we test Wasserstein distance in conjunction with persistent
homology, as a tool for discriminating large scale structures of simulated
universes with different values of $\sigma_8$ cosmological parameter (present
root-mean-square matter fluctuation averaged over a sphere of radius 8 Mpc
comoving). The Wasserstein distance (a.k.a. the pair-matching distance) was
proposed to measure the difference between two networks in terms of persistent
homology. The advantage of this approach consists in its non-parametric way of
probing the topology of the Cosmic web, in contrast to graph-theoretical
approach depending on linking length. By treating the halos of the Cosmic Web
as points in a point cloud we calculate persistent homologies, build
persistence (birth-death) diagrams and evaluate Wasserstein distance between
them. The latter showed itself as a convenient tool to compare simulated Cosmic
webs. We show that one can discern two Cosmic webs (simulated or real) with
different $\sigma_8$ parameter. It turns out that Wasserstein distance's
discrimination ability depends on redshift $z$, as well as on the
dimensionality of considered homology features. We find that the highest
discriminating power this tool obtains at $z=2$ snapshots, among the considered
$z=2$, $1$, and $0.1$ ones.
