  Using a set of $\Lambda$CDM simulations of cosmic structure formation, we
study the evolving connectivity and changing topological structure of the
cosmic web using state-of-the-art tools of multiscale topological data analysis
(TDA). We follow the development of the cosmic web topology in terms of the
evolution of Betti number curves and feature persistence diagrams of the three
(topological) classes of structural features: matter concentrations, filaments
and tunnels, and voids. The Betti curves specify the prominence of features as
a function of density level, and their evolution with cosmic epoch reflects the
changing network connections between these structural features. The persistence
diagrams quantify the longevity and stability of topological features. In this
study we establish, for the first time, the link between persistence diagrams,
the features they show, and the gravitationally driven cosmic structure
formation process. By following the diagrams' development over cosmic time, the
link between the multiscale topology of the cosmic web and the hierarchical
buildup of cosmic structure is established. The sharp apexes in the diagrams
are intimately related to key transitions in the structure formation process.
The apex in the matter concentration diagrams coincides with the density level
at which, typically, they detach from the Hubble expansion and begin to
collapse. At that level many individual islands merge to form the network of
the cosmic web and a large number of filaments and tunnels emerge to establish
its connecting bridges. The location trends of the apex possess a self-similar
character that can be related to the cosmic web's hierarchical buildup. We find
that persistence diagrams provide a significantly higher and more profound
level of information on the structure formation process than more global
summary statistics like Euler characteristic or Betti numbers.
