Targeted Sampling from Massive Block Model Graphs with Personalized PageRank

Abstract

Summary 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.

Statistics in everyone's backyard: an impact study via citation network analysis

“…Here t is a chosen threshold constant. We extend the setting of a single seed node in [14] and [16] to multiple seed nodes, but still make the assumption that they all belong to the same community. While it is unlikely that all papers cited by a specific topic come from the same community, the threshold t helps us prune the vector π and make the assumption more reasonable.…”

“…where p i is the ith entry in the PPR vector. [16] showed that under the DC-SBM, adjusting by the degrees leads to a consistent ordering of the entries in p * so that entries with the highest values belong to the target community. Formally, let n be a community size cutoff.…”

“…Corollary 1 in Chen et al [16] shows that with v 0 = 1 (assuming without loss of generality it belongs to block 1), provided we know the correct size cutoff n = n 1 (recall n 1 = |I 1 |), then the aPPR clustering can recover all the nodes in block 1 with high probability, i.e., C n 1 = I 1 . Since our setting makes use of multiple seed nodes, the follow proposition extends their result to better fit our situation.…”

“…In Figure 6, we further illustrate the distributions of these precision and recall values for the case α = 0.15. For all the case studies from the citation data in the next section, we set α = 0.15 as in Chen et al [16].…”

“…[14] showed that PPR corresponds to the optimal linear classifier under a suitable two-block SBM. Using the more general degree-corrected SBM (DC-SBM, [15]), [16] showed that PPR can include high-degree nodes outside the community of interest, while using the adjusted PPR (aPPR) algorithm in [17] can correct the degree bias, achieving consistency in the detection of the target community with high probability.…”