Subspace Projection Methods for Fast Spectral Embeddings of Evolving Graphs

TL;DR

This paper introduces a subspace projection framework for efficiently updating spectral embeddings of evolving graphs, enabling faster computations while maintaining accuracy for dynamic graph analysis tasks.

Contribution

The paper presents a novel Rayleigh-Ritz based algorithmic framework for updating eigenvectors in dynamic graphs, reducing computational complexity compared to existing methods.

Findings

Lower computational and memory complexity demonstrated.

Strong qualitative performance in eigenvector approximation.

Effective in downstream tasks like node clustering.

Abstract

Several graph data mining, signal processing, and machine learning downstream tasks rely on information related to the eigenvectors of the associated adjacency or Laplacian matrix. Classical eigendecomposition methods are powerful when the matrix remains static but cannot be applied to problems where the matrix entries are updated or the number of rows and columns increases frequently. Such scenarios occur routinely in graph analytics when the graph is changing dynamically and either edges and/or nodes are being added and removed. This paper puts forth a new algorithmic framework to update the eigenvectors associated with the leading eigenvalues of an initial adjacency or Laplacian matrix as the graph evolves dynamically. The proposed algorithm is based on Rayleigh-Ritz projections, in which the original eigenvalue problem is projected onto a restricted subspace which ideally encapsulates the invariant subspace associated with the sought eigenvectors. Following ideas from eigenvector perturbation analysis, we present a new methodology to build the projection subspace. The proposed framework features lower computational and memory complexity with respect to competitive alternatives while empirical results show strong qualitative performance, both in terms of eigenvector approximation and accuracy of downstream learning tasks of central node identification and node clustering.