  Real networks often have severe degree heterogeneity, with the maximum,
average, and minimum node degrees differing significantly. This paper examines
the impact of degree heterogeneity on statistical limits of network data
analysis. Introducing the heterogeneity distribution (HD) under a
degree-corrected mixed-membership network model, we show that the optimal rate
of mixed membership estimation is an explicit functional of the HD. This result
confirms that severe degree heterogeneity may decelerate the error rate, even
when the overall sparsity remains unchanged.
  To obtain a rate-optimal method, we modify an existing spectral algorithm,
Mixed-SCORE, by adding a pre-PCA normalization step. This step normalizes the
adjacency matrix by a diagonal matrix consisting of the $b$th power of node
degrees, for some $b\in \mathbb{R}$. We discover that $b = 1/2$ is universally
favorable. The resulting spectral algorithm is rate-optimal for networks with
arbitrary degree heterogeneity. A technical component in our proofs is
entry-wise eigenvector analysis of the normalized graph Laplacian.
