Subspace Projection Methods for Fast Spectral Embeddings of Evolving Graphs

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.

Figures (6)

• Figure 1: Graph evolution from timestep $t$ to $t+1$. The updated graph $\mathcal{G}^{(t+1)}$ features edge deletions (dashed), edge additions (bold), and newly introduced vertices. In this example: $[\mathbf{K}^{(t+1)}]_{1,3} = [\mathbf{K}^{(t+1)}]_{3,1} = 1$, $[\mathbf{K}^{(t+1)}]_{3,5} = [\mathbf{K}^{(t+1)}]_{5,3} = 1$, $[\mathbf{K}^{(t+1)}]_{2,5} = [\mathbf{K}^{(t+1)}]_{5,2} = -1$, $[\mathbf{G}^{(t+1)}]_{3,1} = 1$, $[\mathbf{G}^{(t+1)}]_{4,1} = 1$, $[\mathbf{G}^{(t+1)}]_{4,2} = 1$, and $[\mathbf{G}^{(t+1)}]_{5,2} = 1$.
• Figure 2: Eigenvector approximation results for dynamic graphs constructed from static graphs (Scenario 1). (a). shows the average of $\psi_{i,t}$ over time for the first three leading eigenvectors. (b). shows the average of $\psi_{i,t}$ over the first $32$ leading eigenvectors as a function of $t$. Results for TIMERS are not reported for the Twitch dataset due to its high time requirement.
• Figure 3: Eigenvector approximation results for graphs with timestampted edges (Scenario 2). (a). shows the average of $\psi_{i,t}$ over time for the first three leading eigenvectors. (b). shows the average of $\psi_{i,t}$ over the first $32$ leading eigenvectors as a function of $t$. For some datasets in (b), insets zoom into the performance of the proposed methods to showcase their differences.
• Figure 4: Run times of different algorithms in seconds for the scenaria of Section \ref{['ssec:eigs-est-results']}. (a). reports the times needed to process datasets of Scenario 1, while (b). reports the times for Scenario 2.
• Figure 5: Effect of $L$ and $P$ parameters on eigenvector approximation and run time of $\textrm{G-REST}_{\rm RSVD}$, compared to $\textrm{G-REST}_3$. (a). plots the performance difference between $\textrm{G-REST}_{\rm RSVD}$ and $\textrm{G-REST}_3$, whose performance is $0.7 \cdot 10^{-2}$. (b). plots the ratio of $\textrm{G-REST}_3$'s run time to that of $\textrm{G-REST}_{\rm RSVD}$, i.e., how many times $\textrm{G-REST}_{\rm RSVD}$ is faster than $\textrm{G-REST}_3$.

Theorems & Definitions (10)

• Proposition 1
• proof
• Corollary 2
• proof
• Theorem 3: Theorem 7.1 demmel1997applied
• Proposition 4
• proof
• Proposition 5
• proof
• Remark 1