  We present a study of the evolution of cosmic filaments across redshift with
an emphasis on some important properties: filament lengths, growth rates, and
radial profiles of galaxy densities. Following an observation-driven approach,
we build cosmic filament catalogues at z=0,1,2,3, and 4 from the galaxy
distributions of the large hydro-dynamical run of the MilleniumTNG project. We
employ the extensively used DisPerSE cosmic web finder code, for which we
provide a user-friendly guide, including the details of a physics-driven
calibration procedure, with the hope of helping future users. We perform the
first statistical measurements of the evolution of connectivity in a
large-scale simulation, finding that the connectivity of cosmic nodes (defined
as the number of filaments attached) globally decreases from early to late
times. The study of cosmic filaments in proper coordinates reveals that
filaments grow in length and radial extent, as expected from large-scale
structures in an expanding Universe. But the most interesting results arise
once the Hubble flow is factored out. We find remarkably stable comoving
filament length functions and over-density profiles, showing only little
evolution of the total population of filaments in the past ~12.25 Gyrs.
However, by tracking the spatial evolution of individual structures, we
demonstrate that filaments of different lengths actually follow different
evolutionary paths. While short filaments preferentially contract, long
filaments expand along their longitudinal direction with growth rates that are
the highest in the early, matter-dominated Universe. Filament diversity at
fixed redshift is also shown by the different (~$5 \sigma$) density values
between the shortest and longest filaments. Our results hint that cosmic
filaments can be used as additional probes for dark energy, but further
theoretical work is still needed.
