🤖 AI Summary
This work addresses the challenge of efficiently computing leading eigenvectors in dynamic graphs, where frequent updates to the adjacency or Laplacian matrix render traditional eigendecomposition methods computationally prohibitive. To overcome this limitation, the authors propose a fast spectral embedding update framework based on Rayleigh-Ritz projection. By leveraging eigenvector perturbation analysis, the method constructs a low-dimensional approximate invariant subspace that preserves high approximation accuracy while substantially reducing computational and memory costs. Experimental results demonstrate that the proposed approach outperforms existing techniques in both the quality of leading eigenvector approximation and performance on downstream tasks—such as influential node identification and node clustering—offering a compelling balance between efficiency and accuracy.
📝 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.