Incremental eigenpair computation for graph Laplacian matrices: theory and applications

Abstract

The smallest eigenvalues and the associated eigenvectors (i.e., eigenpairs) of a graph Laplacian matrix have been widely used in spectral clustering and community detection. However, in real-life applications the number of clusters or communities (say, K) is generally unknown a-priori. Consequently, the majority of the existing methods either choose K heuristically or they repeat the clustering method with different choices of K and accept the best clustering result. The first option, more often, yields suboptimal result, while the second option is computationally expensive. In this work, we propose an incremental method for constructing the eigenspectrum of the graph Laplacian matrix. This method leverages the eigenstructure of graph Laplacian matrix to obtain the K-th smallest eigenpair of the Laplacian matrix given a collection of all previously computed K − 1 smallest eigenpairs. Our proposed method adapts the Laplacian matrix such that the batch eigenvalue decomposition problem transforms into an efficient sequential leading eigenpair computation problem. As a practical application, we consider user-guided spectral clustering. Specifically, we demonstrate that users can utilize the proposed incremental method for effective eigenpair computation and for determining the desired number of clusters based on multiple clustering metrics.

Network community partition based on intelligent clustering algorithm

“…Similar nodes exist in the same community, and there are many connections between nodes in the same community. At present, there are many methods about community division, including graph segmentation [9], label propagation [10], hierarchical clustering [11], matrix spectrum [12], etc.…”

Backward Stability of Explicit External Deflation for the Symmetric Eigenvalue Problem

“…Since the low-rank update of the original A can be done outside of the eigensolver, we refer to this strategy as an explicit external deflation (EED) procedure, to distinguish from the deflation techniques used inside an eigensolver, such as TRLan [20] and ARPACK [9]. Recently the EED has been combined with Krylov subspace methods to progressively compute the eigenpairs toward the interior of the spectrum of symmetric eigenvalue problems [12,21], and extended to structured eigenvalue problems [2,3].…”

A scalable attribute-aware network embedding system

“…BlogCatalog and Flickr are two real-world social media datasets, which are used in [13] 2 . Cora is a dataset based on citations between scientific papers 3 protein-protein interaction (PPI) graphs 4 . The detailed information of these datasets is shown in Table 1.…”

“…As such joint representation not only preserves topology structure, but also has the ability to embeds vertex attributes, edge attributes and other network related information as well. Thus, with the fixed-size and representative embedding vectors, conventional vector-based machine learning algorithms can be naturally introduced to solve diverse problems of analysis on the network, such as node classification [1,16], link prediction [7,9,28,33], node clustering [4,5,30], name disambiguation [25,32,34,35,37,38], and visualization [19,26], etc.…”