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A Local Spectral Method for Graphs: With Applications to Improving Graph Partitions and Exploring Data Graphs Locally
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BibTeX
@MISC{Mahoney_alocal,
author = {Michael W. Mahoney and Lorenzo Orecchia and Nisheeth K. Vishnoi and C○ Michael W. Mahoney and Lorenzo Orecchia and Nisheeth K. Vishnoi Mahoney},
title = {A Local Spectral Method for Graphs: With Applications to Improving Graph Partitions and Exploring Data Graphs Locally},
year = {}
}
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Abstract
The second eigenvalue of the Laplacian matrix and its associated eigenvector are fundamental features of an undirected graph, and as such they have found widespread use in scientific computing, machine learning, and data analysis. In many applications, however, graphs that arise have several local regions of interest, and the second eigenvector will typically fail to provide information fine-tuned to each local region. In this paper, we introduce a locally-biased analogue of the second eigenvector, and we demonstrate its usefulness at highlighting local properties of data graphs in a semi-supervised manner. To do so, we first view the second eigenvector as the solution to a constrained optimization problem, and we incorporate the local information as an additional constraint; we then characterize the optimal solution to this new problem and show that it can be interpreted as a generalization of a Personalized PageRank vector; and finally, as a consequence, we show that the solution can be computed in nearly-linear time. In addition, we show that this locally-biased vector can be used to compute an approximation to the best partition near an input seed set in a manner analogous to the way in which the second eigenvector of the Laplacian can be used to obtain an approximation to the best partition in the entire input graph. Such a primitive is useful
Keyphrases
second eigenvector exploring data graph locally improving graph partition local spectral method many application new problem local property locally-biased analogue input seed local information local region nearly-linear time undirected graph optimal solution scientific computing laplacian matrix locally-biased vector widespread use data analysis entire input graph several local region fundamental feature optimization problem personalized pagerank vector second eigenvalue semi-supervised manner machine learning additional constraint data graph
