Block Models and Personalized PageRank (1607.03483v1)

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.

• The paper establishes a theoretical link between stochastic block models and personalized PageRank, proving their optimal equivalence under specific conditions.
• It derives the optimal gradient for distinguishing landing probabilities, thereby justifying PPR’s effectiveness in seed set expansion tasks.
• The research introduces enhanced scoring techniques, leveraging higher moments to rival belief propagation in community recovery performance.

An Academic Overview of "Block Models and Personalized PageRank"

This paper, authored by Isabel Kloumann, Johan Ugander, and Jon Kleinberg, presents an investigation into the seed set expansion problem utilizing the personalized PageRank (PPR) algorithm, set within the framework of stochastic block models (SBMs). The seed set expansion problem challenges researchers to identify communities within a network given only a small subset of nodes from the community of interest. This problem is significant in network structure analysis, with applications spanning social networks, web analysis, and community detection.

The work details how standard node ranking techniques, such as PPR and heat kernels, are generally applied to landing probabilities derived from random walks. These methods have been successful in various practical applications but lack a formal relationship to the seed set expansion problem's specific objectives. This research addresses this gap by contextualizing these techniques within the SBM framework, enabling a principled evaluation and improvement of these methods.

Contributions

1. Theoretical Framework: The authors establish a connection between stochastic block models and personalized PageRank, demonstrating that under certain reasonable assumptions, the optimal solution in the space of landing probabilities is equivalent to PPR with particular parameters. This is a notable theoretical discovery, as it provides a rigorous basis for understanding why PPR performs well in seed set expansion tasks.
2. Optimal Gradient Derivation: The work derives the optimal gradient for separating landing probabilities of different classes within SBMs. This derivation shows that for specific parameter choices, it aligns with PPR values, thus providing justification for PPR's effectiveness in ranking nodes and expanding seed sets.
3. Improved Techniques: The research proposes advanced scoring techniques that incorporate higher moments of landing probabilities. These methods, though implemented as linear classification rules, significantly enhance performance and show competitiveness with belief propagation methods.
4. Geometric and Fisherian Discriminant Functions: Beyond simple linear models, the paper considers more complex approaches like Fisherian discriminant functions that can account for variance and covariance in landing probabilities, thus improving classification accuracy in practical applications.

Numerical Results

The paper supports its claims with robust numerical results. These include enhanced performance by the proposed methods over traditional PPR and heat kernels in recovering seed communities from graphs generated by SBMs. The proposed techniques also demonstrate performance comparable to belief propagation, a known optimal but computationally complex method.

Implications and Future Directions

This research offers several impactful implications:

• By establishing a formal foundation connecting stochastic block models to personalized PageRank, the research potentially broadens the applicability of PPR to more general community detection and ranking challenges across diverse datasets.
• The framework and methods proposed can be adapted to refine other graph-based algorithms, possibly enhancing performance in unsupervised settings where community labels are unknown.
• Future work could explore applying these insights to develop alternative random-walk models, such as non-backtracking walks, or other graph models where traditional methods are less effective.
• The principles outlined could inspire the development of new machine learning approaches within structured models, broadening the scope and fidelity of automatic community detection techniques.

This paper provides a rigorous, mathematical perspective on the seed set expansion problem, with significant contributions to both theory and practice. Its findings highlight critical connections and optimizations that could influence future research in network analysis and machine learning applied to graph data.