Block models and personalized PageRank

Abstract

Methods for ranking the importance of nodes in a network have a rich history in machine learning and across domains that analyze structured data. Recent work has evaluated these methods through the "seed set expansion problem": given a subset S of nodes from a community of interest in an underlying graph, can we reliably identify the rest of the community? We start from the observation that the most widely used techniques for this problem, personalized PageRank and heat kernel methods, operate in the space of "landing probabilities" of a random walk rooted at the seed set, ranking nodes according to weighted sums of landing probabilities of different length walks. Both schemes, however, lack an a priori relationship to the seed set objective. In this work, we develop a principled framework for evaluating ranking methods by studying seed set expansion applied to the stochastic block model. We derive the optimal gradient for separating the landing probabilities of two classes in a stochastic block model and find, surprisingly, that under reasonable assumptions the gradient is asymptotically equivalent to personalized PageRank for a specific choice of the PageRank parameter α that depends on the block model parameters. This connection provides a formal motivation for the success of personalized PageRank in seed set expansion and node ranking generally. We use this connection to propose more advanced techniques incorporating higher moments of landing probabilities; our advanced methods exhibit greatly improved performance, despite being simple linear classification rules, and are even competitive with belief propagation.PageRank | stochastic block models | seed set expansion T he challenge of contextually ranking nodes in a network has emerged as a problem of canonical significance in many domains, with a particularly rich history of study in social and information networks (1-4). An active line of recent work has focused on the problem of "seed set expansion" in networks (5-11), a fundamental version of node ranking with the following natural definition.In the seed set expansion problem, we are given a graph G representing some form of social or information network, and there is a hidden community of interest that we would like to find, corresponding to an internally well-connected set of nodes. We know a small subset S of the nodes in this community, and from this "seed set" S , we would like to expand outward to find the rest of the community-by ordering the rest of the nodes outside S according to some ranking criterion and proposing nodes in this order as additional members of the community. This problem arises in a wide range of domains, including settings where we are trying to find web pages that are related to a set of examples, to identify a social group from a set of sample members provided by a domain expert, or to help a user automatically populate a group they are defining in an online social-networking application.A recent focus in the work on this problem has been the power of approaches based on ran...

Targeted Sampling from Massive Block Model Graphs with Personalized PageRank

“…Breaking the symmetry that was imposed by Kloumann et al . () reveals additional insight. In particular, given a seed node in the first block, we show that PPR is likely to contain high degree nodes outside that block.…”

“…Kloumann et al . () used such a scenario. As a by‐product of our analysis, we extend their results under the DCSBM with the symmetric conditions (see the on‐line supplementary materials section S3 to the paper).…”

Local Community Detection in Dynamic Graphs Using Personalized Centrality

“…The entire personalized PageRank matrix, formed with each column starting from the corresponding vertex, has been shown asymptotically to recover the stochastic block model used here as a test case [49].…”

High-Order Degree and Combined Degree in Complex Networks

“…Network eigenvector centrality [33,34] is often used to describe the importance of nodes in social network. In 1998, Brin and Page [35,36] simplified the eigenvector centrality for undirected network into PageRank algorithm ( ), which is widely used in searching engine Google [36] and many other directed networks [37][38][39][40][41][42]. By Choose the origin node 1: (a) the process of splitting and random walking could be repeated an infinite number of times, because origin node 1 is connected to node 2; (b) the ball can walk randomly an infinite number of times without split since the origin node is one of the directed loop; (c) the ball split and walk randomly only once; (d) the ball can only walk twice without splitting; (e) the ball can walk an infinite number of times without split.…”

“…The computation of both closeness centrality and betweenness centrality are so complicated that big networks often need fast approximate algorithm [46][47][48]. Even though the eigenvector centrality on undirected network and the PageRank on directed work can give satisfying results in nodes sorting, those two methods often involve expensive computations, such as iterations [42]. In the following paragraphs, we will define several novel centrality metrics, which are cheaper in computation compared with closeness centrality, betweenness centrality, eigenvector centrality on undirected network, and PageRank on directed network.…”