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With the halo catalog from the {\it Millennium Simulation}, we analyze theweak-lensing measured density profiles for clusters of galaxies, payingattention to the determination of the concentration-mass ($c$-$M$) relationwhich can be biased by the center offset, selection effect, and shape noisefrom intrinsic ellipticities of background galaxies. Several different methodsof locating the center of a cluster from weak-lensing effects alone areexplored. We find that, for intermediate redshift clusters, the highest peakfrom our newly proposed two-scale smoothing method applied to the reconstructedconvergence field, first with a smoothing scale of $2^\prime$ and then$0\overset{\prime}{.}5$, corresponds best to the true center. Assuming theparameterized Navarro-Frenk-White profile, we fit the reduced tangential shearsignals around different centers identified by different methods. It is shownthat, for the ensemble median values, a center offset larger than one scaleradius $r_s$ can bias the derived mass and concentration significantly lowerthan the true values, especially for low-mass halos. However, the existence ofnoise 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 isinsignificant if an appropriate center finding method is adopted. On the otherhand, noise from intrinsic ellipticities can bias the $c$-$M$ relation derivedfrom a sample of weak-lensing analyzed clusters if a simple $\chi^2$ fittingmethod 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.

Effects of center offset and noise on weak-lensing derived concentration-mass relation of dark matter halos