  The paper provides statistical theory and intuition for personalized PageRank
(called "PPR"): a popular technique that samples a small community from a
massive network. We study a setting where the entire network is expensive to
obtain thoroughly or to maintain, but we can start from a seed node of interest
and "crawl" the network to find other nodes through their connections. By
crawling the graph in a designed way, the PPR vector can be approximated
without querying the entire massive graph, making it an alternative to snowball
sampling. Using the degree-corrected stochastic block model, we study whether
the PPR vector can select nodes that belong to the same block as the seed node.
We provide a simple and interpretable form for the PPR vector, highlighting its
biases towards high degree nodes outside the target block. We examine a simple
adjustment based on node degrees and establish consistency results for PPR
clustering that allows for directed graphs. These results are enabled by recent
technical advances showing the elementwise convergence of eigenvectors. We
illustrate the method with the massive Twitter friendship graph, which we crawl
by using the Twitter application programming interface. We find that the
adjusted and unadjusted PPR techniques are complementary approaches, where the
adjustment makes the results particularly localized around the seed node, and
that the bias adjustment greatly benefits from degree regularization.
