  Consider an undirected network with $n$ nodes and $K$ perceivable
communities, where some nodes may have mixed memberships. We assume that for
each node $1 \leq i \leq n$, there is a probability mass function $\pi_i$
defined over $\{1, 2, \ldots, K\}$ such that \[ \pi_i(k) = \mbox{the weight of
node $i$ on community $k$}, \qquad 1 \leq k \leq K. \] The goal is to estimate
$\{\pi_i, 1 \leq i \leq n\}$ (i.e., membership estimation).
  We model the network with the {\it degree-corrected mixed membership (DCMM)}
model \cite{Mixed-SCORE}. Since for many natural networks, the degrees have an
approximate power-law tail, we allow {\it severe degree heterogeneity} in our
model.
  For any membership estimation $\{\hat{\pi}_i, 1 \leq i \leq n\}$, since each
$\pi_i$ is a probability mass function, it is natural to measure the errors by
the average $\ell^1$-norm \[ \frac{1}{n} \sum_{i = 1}^n \| \hat{\pi}_i -
\pi_i\|_1. \] We also consider a variant of the $\ell^1$-loss, where each
$\|\hat{\pi}_i - \pi_i\|_1$ is re-weighted by the degree parameter $\theta_i$
in DCMM (to be introduced).
  We present a sharp lower bound. We also show that such a lower bound is
achievable under a broad situation. More discussion in this vein is continued
in our forthcoming manuscript.
  The results are very different from those on community detection. For
community detection, the focus is on the special case where all $\pi_i$ are
degenerate; the goal is clustering, so Hamming distance is the natural choice
of loss function, and the rate can be exponentially fast. The setting here is
broader and more difficult: it is more natural to use the $\ell^1$-loss, and
the rate is only polynomially fast.
