Incremental eigenpair computation for graph Laplacian matrices: theory and applications (original) (raw)
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The smallest eigenvalues and the associated eigenvectors (i.e., eigenpairs) of a graph Laplacian matrix have been widely used for spectral clustering and community detection. However, in real-life applications the number of clusters or communities (say, KKK) is generally unknown a-priori. Consequently, the majority of the existing methods either choose KKK heuristically or they repeat the clustering method with different choices of KKK 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 KKK-th eigenpairs of the Laplacian matrix given a collection of all the Kâ1K-1Kâ1 smallest eigenpairs. Our proposed method adapts the Laplacian matrix such that the batch eigenvalue decomposition problem transforms int...
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Partitioning a graph into groups of vertices such that those within each group are more densely connected than vertices assigned to different groups, known as graph clustering, is often used to gain insight into the organization of large scale networks and for visualization purposes. Whereas a large number of dedicated techniques have been recently proposed for static graphs, the design of on-line graph clustering methods tailored for evolving networks is a challenging problem, and much less documented in the literature. Motivated by the broad variety of applications concerned, ranging from the study of biological networks to graphs of scientific references through to the exploration of communications networks such as the World Wide Web, it is the main purpose of this paper to introduce a novel, computationally efficient, approach to graph clustering in the evolutionary context. Namely, the method promoted in this article is an incremental eigenvalue solution for the spectral clustering method described by Ng. et al. (2001). Beyond a precise description of its practical implementation and an evaluation of its complexity, its performance is illustrated through numerical experiments, based on datasets modelling the evolution of a HIV epidemic and the purchase history graph of an e-commerce website.
Efficient eigen-updating for spectral graph clustering
Neurocomputing, 2014
Partitioning a graph into groups of vertices such that those within each group are more densely connected than vertices assigned to different groups, known as graph clustering, is often used to gain insight into the organisation of large scale networks and for visualisation purposes. Whereas a large number of dedicated techniques have been recently proposed for static graphs, the design of on-line graph clustering methods tailored for evolving networks is a challenging problem, and much less documented in the literature. Motivated by the broad variety of applications concerned, ranging from the study of biological networks to the analysis of networks of scientific references through the exploration of communications networks such as the World Wide Web, it is the main purpose of this paper to introduce a novel, computationally efficient, approach to graph clustering in the evolutionary context. Namely, the method promoted in this article can be viewed as an incremental eigenvalue solution for the spectral clustering method described by . The incremental eigenvalue solution is a general technique for finding the approximate eigenvectors of a symmetric matrix given a change. As well as outlining the approach in detail, we present a theoretical bound on the quality of the approximate eigenvectors using perturbation theory. We then derive a novel spectral clustering algorithm called Incremental Approximate Spectral Clustering (IASC). The IASC algorithm is simple to implement and its efficacy is demonstrated on both synthetic and real datasets modelling the evolution of a HIV epidemic, a citation network and the purchase history graph of an e-commerce website.
Improvement on Incremental Spectral Clustering
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Spectral clustering (SC) is one of the most popular algorithms for finding communities in a static graph. The algorithm derives the communities through clustering the leading eigenvectors of the graph Laplacian. Many practical problems involve networks that evolve over time (for instance, the Facebook friendship network) and communities need to be updated dynamically. Since spectral clustering is an offline algorithm, any change in the graph requires a new computation of the eigenvectors of the graph Laplacian matrix, which is highly inefficient for real-time clustering of most large dynamic networks. Incremental spectral clustering (ISC) extends the method to efficiently cluster dynamic graphs [1], by incrementally updating the eigenvectors and eigenvalues, but can sometimes lead to sub-optimal performance due to approximation errors. In this paper, a modified incremental spectral algorithm (MISC) is introduced by considering higher-order approximations of the eigenvector update ru...
Clustering Based on Eigenvectors of the Adjacency Matrix
International Journal of Applied Mathematics and Computer Science, 2018
The paper presents a novel spectral algorithm EVSA (eigenvector structure analysis), which uses eigenvalues and eigenvectors of the adjacency matrix in order to discover clusters. Based on matrix perturbation theory and properties of graph spectra we show that the adjacency matrix can be more suitable for partitioning than other Laplacian matrices. The main problem concerning the use of the adjacency matrix is the selection of the appropriate eigenvectors. We thus propose an approach based on analysis of the adjacency matrix spectrum and eigenvector pairwise correlations. Formulated rules and heuristics allow choosing the right eigenvectors representing clusters, i.e., automatically establishing the number of groups. The algorithm requires only one parameter-the number of nearest neighbors. Unlike many other spectral methods, our solution does not need an additional clustering algorithm for final partitioning. We evaluate the proposed approach using real-world datasets of different ...
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