  With the halo catalog from the {\it Millennium Simulation}, we analyze the
weak-lensing measured density profiles for clusters of galaxies, paying
attention to the determination of the concentration-mass ($c$-$M$) relation
which can be biased by the center offset, selection effect, and shape noise
from intrinsic ellipticities of background galaxies. Several different methods
of locating the center of a cluster from weak-lensing effects alone are
explored. We find that, for intermediate redshift clusters, the highest peak
from our newly proposed two-scale smoothing method applied to the reconstructed
convergence field, first with a smoothing scale of $2^\prime$ and then
$0\overset{\prime}{.}5$, corresponds best to the true center. Assuming the
parameterized Navarro-Frenk-White profile, we fit the reduced tangential shear
signals around different centers identified by different methods. It is shown
that, for the ensemble median values, a center offset larger than one scale
radius $r_s$ can bias the derived mass and concentration significantly lower
than the true values, especially for low-mass halos. However, the existence of
noise can compensate for the offset effect and reduce the systematic bias,
although the scatter of mass and concentration becomes considerably larger.
Statistically, the bias effect of center offset on the $c$-$M$ relation is
insignificant if an appropriate center finding method is adopted. On the other
hand, noise from intrinsic ellipticities can bias the $c$-$M$ relation derived
from a sample of weak-lensing analyzed clusters if a simple $\chi^2$ fitting
method is used. To properly account for the scatter and covariance between $c$
and $M$, we apply a Bayesian method to improve the statistical analysis of the
$c$-$M$ relation. It is shown that this new method allows us to derive the
$c$-$M$ relation with significantly reduced biases.
