[Paper Review] Eigenvector Computation and Community Detection in Asynchronous Gossip Models

Frederik Mallmann-Trenn, Cameron Musco+1 more|arXiv (Cornell University)|Apr 23, 2018

Complex Network Analysis Techniques6 citations

TL;DR

This paper presents a novel asynchronous distributed algorithm for computing top eigenvectors of a network's adjacency matrix using a modified version of Oja's streaming PCA algorithm. It achieves state-of-the-art performance in community detection under the stochastic block model and a randomized communication model, with formal convergence guarantees and improved time complexity bounds for eigenvector approximation and community recovery.

ABSTRACT

We give a simple distributed algorithm for computing adjacency matrix eigenvectors for the communication graph in an asynchronous gossip model. We show how to use this algorithm to give state-of-the-art asynchronous community detection algorithms when the communication graph is drawn from the well-studied stochastic block model. Our methods also apply to a natural alternative model of randomized communication, where nodes within a community communicate more frequently than nodes in different communities. Our analysis simplifies and generalizes prior work by forging a connection between asynchronous eigenvector computation and Oja's algorithm for streaming principal component analysis. We hope that our work serves as a starting point for building further connections between the analysis of stochastic iterative methods, like Oja's algorithm, and work on asynchronous and gossip-type algorithms for distributed computation.

Motivation & Objective

• To develop a distributed, asynchronous algorithm for computing top eigenvectors of a network's adjacency matrix in a gossip-based communication model.
• To apply this eigenvector computation to improve community detection in the stochastic block model and related models with intra-community communication bias.
• To establish finite-time convergence guarantees for eigenvector approximation under asynchronous and randomized scheduling.
• To bridge the analysis of streaming iterative methods (like Oja’s algorithm) with asynchronous distributed graph algorithms.
• To simplify and generalize prior work on eigenvector computation and community detection in distributed systems.

Proposed method

• Adapts Oja’s algorithm for streaming PCA to the asynchronous gossip model, enabling distributed computation of top-k eigenvectors.
• Each node maintains and updates local estimates of the eigenvectors using messages from randomly selected neighbors.
• Uses a normalized adjacency matrix representation to ensure stability and convergence in the asynchronous setting.
• Employs Chernoff bounds and martingale concentration to analyze label propagation and convergence under random communication schedules.
• Introduces a filtering mechanism to reduce error propagation by focusing on high-confidence label updates.
• Leverages spectral gap and eigenvalue gap parameters to bound convergence time and error rates.

Experimental results

Research questions

• RQ1 Can Oja’s streaming PCA algorithm be adapted to work in fully asynchronous, randomized communication models like the gossip model?
• RQ2 What are the convergence time and error bounds for distributed eigenvector computation in such models?
• RQ3 How does the algorithm perform in community detection tasks under the stochastic block model with asynchronous communication?
• RQ4 Can the analysis of streaming eigenvector estimation be extended to analyze asynchronous distributed graph algorithms?
• RQ5 What is the impact of spectral gap and mixing time on the convergence rate of the proposed algorithm?

Key findings

• The algorithm computes the top-k eigenvectors with error at most ε in time Õ(Λk³/(gap·min(gap, γ_mix)·ε³)), where Λ is the sum of the k largest eigenvalues.
• For the stochastic block model G(n,p,q), the algorithm achieves community detection with high probability when p = Ω(log n / n) and q = Ω(log n / n), with a time complexity of Õ(n log n) rounds.
• The method ensures that the number of mislabeled nodes decreases geometrically over time, with |S_{t+1}| ≤ (2/3)|S_t| w.h.p. after sufficient rounds.
• The analysis shows that the algorithm converges to the correct eigenvectors even under asynchronous and randomized communication, with strong finite-time guarantees.
• The algorithm improves upon prior work in both convergence time and error bounds for distributed community detection in the stochastic block model.
• The connection between Oja’s algorithm and asynchronous gossip models enables a unified analysis framework for distributed eigenvector computation.