A fast parametric maximum flow algorithm and applications
SIAM Journal on Computing
An algorithm for drawing general undirected graphs
Information Processing Letters
SIAM Journal on Scientific and Statistical Computing
The anatomy of a large-scale hypertextual Web search engine
WWW7 Proceedings of the seventh international conference on World Wide Web 7
Normalized Cuts and Image Segmentation
IEEE Transactions on Pattern Analysis and Machine Intelligence
Scaling personalized web search
WWW '03 Proceedings of the 12th international conference on World Wide Web
Topic-Sensitive PageRank: A Context-Sensitive Ranking Algorithm for Web Search
IEEE Transactions on Knowledge and Data Engineering
Convex Optimization
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Graph partitioning using single commodity flows
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Proceedings of the 15th international conference on World Wide Web
Local Graph Partitioning using PageRank Vectors
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
An algorithm for improving graph partitions
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Statistical properties of community structure in large social and information networks
Proceedings of the 17th international conference on World Wide Web
On partitioning graphs via single commodity flows
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Conductance and convergence of Markov chains-a combinatorial treatment of expanders
SFCS '89 Proceedings of the 30th Annual Symposium on Foundations of Computer Science
Empirical comparison of algorithms for network community detection
Proceedings of the 19th international conference on World wide web
Detecting people using mutually consistent poselet activations
ECCV'10 Proceedings of the 11th European conference on Computer vision: Part VI
Approaching Optimality for Solving SDD Linear Systems
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
CVPR '11 Proceedings of the 2011 IEEE Conference on Computer Vision and Pattern Recognition
Approximate computation and implicit regularization for very large-scale data analysis
PODS '12 Proceedings of the 31st symposium on Principles of Database Systems
Overlapping community detection using seed set expansion
Proceedings of the 22nd ACM international conference on Conference on information & knowledge management
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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 for identifying and refining clusters locally, as it allows us to focus on a local region of interest in a semi-supervised manner. Finally, we provide a detailed empirical evaluation of our method by showing how it can applied to finding locally-biased sparse cuts around an input vertex seed set in social and information networks.