English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

The second eigenvalue of the Laplacian matrix and its associated eigenvectorare fundamental features of an undirected graph, and as such they have foundwidespread use in scientific computing, machine learning, and data analysis. Inmany applications, however, graphs that arise have several \emph{local} regionsof interest, and the second eigenvector will typically fail to provideinformation fine-tuned to each local region. In this paper, we introduce alocally-biased analogue of the second eigenvector, and we demonstrate itsusefulness at highlighting local properties of data graphs in a semi-supervisedmanner. To do so, we first view the second eigenvector as the solution to aconstrained optimization problem, and we incorporate the local information asan additional constraint; we then characterize the optimal solution to this newproblem and show that it can be interpreted as a generalization of aPersonalized PageRank vector; and finally, as a consequence, we show that thesolution can be computed in nearly-linear time. In addition, we show that thislocally-biased vector can be used to compute an approximation to the bestpartition \emph{near} an input seed set in a manner analogous to the way inwhich the second eigenvector of the Laplacian can be used to obtain anapproximation to the best partition in the entire input graph. Such a primitiveis useful for identifying and refining clusters locally, as it allows us tofocus on a local region of interest in a semi-supervised manner. Finally, weprovide a detailed empirical evaluation of our method by showing how it canapplied to finding locally-biased sparse cuts around an input vertex seed setin social and information networks.

A Local Spectral Method for Graphs: with Applications to Improving Graph Partitions and Exploring Data Graphs Locally