Block Models and Personalized PageRank

TL;DR

This paper provides a theoretical foundation for personalized PageRank's effectiveness in seed set expansion by linking it to optimal gradients in stochastic block models, and introduces improved methods leveraging higher moments.

Contribution

It develops a principled framework connecting personalized PageRank to optimal seed set expansion, and proposes advanced techniques with enhanced performance.

Findings

Personalized PageRank is asymptotically equivalent to the optimal gradient in stochastic block models.

Higher-moment based methods significantly outperform basic personalized PageRank.

The proposed methods are competitive with belief propagation.

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 though 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 $\alpha$ that depends on the block model parameters. This connection provides a novel 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.