  In the regime of cosmology and large-scale structure formation, filaments are
vital components of the cosmic web. This study employs statistical methods to
examine the formation, evolution, and cosmological constraints of filaments
identified by DisPerSe. We run large-sample of N-body simulations to study the
filament length and mass functions and their evolution. In general, the
filament length function can be fitted by a power law with both the
normalization and power index dependent on redshift and cosmological
parameters. It is discovered that filament properties are influenced by various
cosmological parameters, with $\sigma_8$ and $n_s$ exhibiting slightly stronger
dependence than $\Omega_m$. We also uncover a three-stage filament formation
process from $z \sim 3$ to $z \sim 1$: rapid formation of long and short
filaments from $z \sim 3$ to $z \sim 2$, persistence of long filaments from $z
\sim 2$ to $z \sim 1$, followed by fragmentation and increased prevalence of
shorter filaments below $z \sim 1$. Additionally, both large and small-scale
perturbations in power spectrum influence filament growth, with a
characteristic scale around $k \sim 0.265$, wherein changes in matter
fluctuations larger than this scale result in the formation of long filaments,
while changes smaller than this scale have the opposite effect. These insights
enhance our understanding of filament evolution and their cosmological
relevance and also highlight their potential cosmological applications in
observations.
