  In the context of the cosmological and constrained ELUCID simulation, this
study explores the statistical characteristics of filaments within the cosmic
web, focussing on aspects such as the distribution of filament lengths and
their radial density profiles. Using the classification of the cosmic web
environment through the Hessian matrix of the density field, our primary focus
is on how cosmic structures react to the two variables $R_{\rm s}$ and
$\lambda_{\rm th}$. The findings show that the volume fractions of knots,
filaments, sheets, and voids are highly influenced by the threshold parameter
$\lambda_{\rm th}$, with only a slight influence from the smoothing length
$R_{\rm s}$. The central axis of the cylindrical filament is pinpointed using
the medial-axis thinning algorithm of the COWS method. It is observed that
median filament lengths tend to increase as the smoothing lengths increase.
Analysis of filament length functions at different values of $R_{\rm s}$
indicates a reduction in shorter filaments and an increase in longer filaments
as $R_{\rm s}$ increases, peaking around $2.5R_{\rm s}$. The study also shows
that the radial density profiles of filaments are markedly affected by the
parameters $R_{\rm s}$ and $\lambda_{\rm th}$, showing a valley at
approximately $2R_{\rm s}$, with increases in the threshold leading to higher
amplitudes of the density profile. Moreover, shorter filaments tend to have
denser profiles than their longer counterparts.
