[Paper Review] Community Detection Based on the $L_\infty$ convergence of eigenvectors in DCBM

Yan Liu, Zhiqiang Hou+4 more|arXiv (Cornell University)|Jun 16, 2019

Complex Network Analysis Techniques33 references4 citations

TL;DR

This paper establishes the $L_\infty$ convergence rate of eigenvectors for the adjacency matrix under the degree-corrected stochastic block model (DCBM), proving almost sure convergence at rate $o_{a.s.}(n^{-1}\log n)$. It introduces a novel spectral clustering method, SCDRE, based on the ratio of eigenvector components, which achieves consistent community detection and outperforms existing methods in simulations.

ABSTRACT

Spectral clustering is one of the most popular algorithms for community detection in network analysis. Based on this rationale, in this paper we give the convergence rate of eigenvectors for the adjacency matrix in the $l_\infty$ norm, under the stochastic block model (BM) and degree corrected stochastic block model (DCBM), adding some mild and rational conditions. We also extend this result to a more general model, presented based on the DCBM such that the value of random variables in the adjacency matrix is not 0 or 1, but an arbitrary real number. During the process of proving the above conclusion, we obtain the relationship of the eigenvalues in the adjacency matrix and the corresponding `population' matrix, which vary in dimension from the community-wise edge probability matrix. Using that result, we can give an estimate of the number of the communities in a known set of network data. Meanwhile we proved the consistency of the estimator. Furthermore, according to the derivation of proof for the convergence of eigenvectors, we propose a new approach to community detection -- Spectral Clustering based on Difference of Ratios of Eigenvectors (SCDRE). Our simulation experiments demonstrate the superiority of our method in community detection.

Motivation & Objective

• To establish the almost sure $L_\infty$ convergence rate of eigenvectors for the adjacency matrix under the DCBM.
• To derive the relationship between eigenvalues of the adjacency matrix and the corresponding population matrix in the DCBM framework.
• To develop a new community detection method, SCDRE, based on the ratio of eigenvector components.
• To provide a consistent estimator for the number of communities using eigenvalue and eigenvector relationships.
• To extend theoretical results to a generalized model where adjacency matrix entries are arbitrary real numbers, not just 0 or 1.

Proposed method

• Uses random matrix theory and Wigner matrix results to analyze eigenvector convergence in the $L_\infty$ norm.
• Derives the convergence rate $\|U_t - Q_t T_t\|_\infty = o_{a.s.}(n^{-1}\log n)$ for leading eigenvectors under mild conditions.
• Introduces the Spectral Clustering based on Difference of Ratios of Eigenvectors (SCDRE) method using ratios of eigenvector components.
• Establishes consistency of the community number estimator via eigenvalue and eigenvector structure in the population matrix.
• Extends theoretical results to a generalized model where adjacency matrix entries are arbitrary real numbers, not restricted to 0–1.
• Applies the Davis–Kahan theorem and orthogonal decomposition techniques to bound eigenvector perturbations in $L_\infty$.

Experimental results

Research questions

• RQ1 What is the almost sure $L_\infty$ convergence rate of eigenvectors for the adjacency matrix under the DCBM?
• RQ2 How do the eigenvalues and eigenvectors of the adjacency matrix relate to those of the population matrix in the DCBM?
• RQ3 Can the ratio of eigenvector components be used to consistently estimate the number of communities in a network?
• RQ4 Does the proposed SCDRE method achieve better community detection performance than existing spectral clustering methods?
• RQ5 Can the theoretical convergence results be extended to a generalized model with arbitrary real-valued adjacency entries?

Key findings

• The eigenvectors of the adjacency matrix converge to those of the population matrix in the $L_\infty$ norm at rate $o_{a.s.}(n^{-1}\log n)$ under the DCBM.
• The proposed SCDRE method achieves superior community detection performance in simulation experiments compared to baseline spectral clustering.
• The number of communities can be consistently estimated using the relationship between eigenvalues and eigenvectors of the adjacency and population matrices.
• The eigenvector ratios $RR_\zeta(i,j)$ are shown to be 1 only if nodes $i$ and $j$ are in the same community, with a contradiction arising if they are in different communities.
• Theoretical consistency is proven for the community number estimator under the DCBM with mild regularity conditions.
• The convergence results are extended to a generalized model where adjacency matrix entries are arbitrary real numbers, not limited to 0 or 1.