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## A local clustering algorithm for massive graphs and its application to nearly-linear time graph partitioning (2013)

Citations: | 57 - 8 self |

### Citations

3176 | The PageRank Citation Ranking: Bringing Order to the Web
- Page, Brin, et al.
- 1999
(Show Context)
Citation Context ...e produces is local to the starting vertex v in the sense that it consists of nodes that are among the most favored destinations of random walks starting from v. By using the personal PageRank vector =-=[PBMW98]-=- to define nearness, Andersen, Chung, and Lang [ACL06] have produced an improved version of our algorithm Nibble, which they call PageRank-Nibble. Their algorithm is further improved by Andersen and P... |

352 | Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms
- Leighton, Rao
- 1999
(Show Context)
Citation Context ...as an optimization problem: Given an undirected graph G and a parameter φ, find a cluster C such that Φ (C) ≤ φ or determine that no such cluster exists. The problem is NP-complete (see, for example, =-=[LR99]-=- or [SS06]). But, approximation algorithms exist. Leighton and Rao [LR99] used linear programming to obtainO(log n)-approximations of the sparsest cut. Arora, Rao, and Vazirani [ARV04] improved this t... |

333 |
A lower bound for the smallest eigenvalue of the Laplacian, in “Problems in analysis (Papers dedicated to Salomon Bochner,1969
- Cheeger
- 1970
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Citation Context ...nd ψS is the restriction of that walk to the set S. 2.1.2. Vector-based graph partitioning. Like the spectral partitioning algorithm of Hagen and Kahng [HK92] and that implicit in the work of Cheeger =-=[Che70]-=- (see also [Mih89, ST96]), and the probability-based algorithm implicit in the work of Lovász and Simonovits [LS90], Nibble extracts a cluster from a vector by the following algorithm: 1. Given an n-... |

311 | Expander flows, geometric embeddings and graph partitioning
- Arora, Rao, et al.
(Show Context)
Citation Context ...ee, for example, [LR99] or [SS06]). But, approximation algorithms exist. Leighton and Rao [LR99] used linear programming to obtainO(log n)-approximations of the sparsest cut. Arora, Rao, and Vazirani =-=[ARV04]-=- improved this to O( √ logn) through semidefinite programming. Faster algorithms obtaining similar guarantees have been constructed by Arora, Hazan, and Kale [AHK04], Khandekar, Rao, and Vazirani [KRV... |

294 | New spectral methods for ratio cut partitioning and clustering
- Hagen, Kahng
- 1992
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Citation Context ... steady-state distribution of any random walk, and ψS is the restriction of that walk to the set S. 2.1.2. Vector-based graph partitioning. Like the spectral partitioning algorithm of Hagen and Kahng =-=[HK92]-=- and that implicit in the work of Cheeger [Che70] (see also [Mih89, ST96]), and the probability-based algorithm implicit in the work of Lovász and Simonovits [LS90], Nibble extracts a cluster from a ... |

220 | Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems
- Spielman, Teng
- 2004
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Citation Context ... first in a sequence of three papers expanding on material that appeared first under the title “Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems” =-=[ST04]-=-. The second paper, “Spectral sparsification of graphs” [ST11], contains further results on partitioning graphs and applies them to producing spectral sparsifiers of graphs. The third paper, “Nearly-l... |

196 | Local graph partitioning using pagerank vectors. Pages 475–486 of: Foundations of computer science, 2006. focs’06. 47th annual ieee symposium on
- Andersen, Chung, et al.
- 2006
(Show Context)
Citation Context ...se that it consists of nodes that are among the most favored destinations of random walks starting from v. By using the personal PageRank vector [PBMW98] to define nearness, Andersen, Chung, and Lang =-=[ACL06]-=- have produced an improved version of our algorithm Nibble, which they call PageRank-Nibble. Their algorithm is further improved by Andersen and Peres [AP09], who base their algorithm on the volume-bi... |

196 | Spectral partitioning works: Planar graphs and finite element meshes - Spielman, Teng - 1996 |

161 |
Random walks in a convex body and an improved volume algorithm
- Lovász, Simonovits
- 1993
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Citation Context ...ndom walks from a vertex in S, without reference to the truncation we perform in the algorithm. We then analyze the impact of truncation used in Nibble and extend the theory of Lovász and Simonovits =-=[LS93]-=- to truncated random walks. 12 DANIEL A. SPIELMAN AND SHANG-HUA TENG Step 1: The diffusion core of a cluster and its properties. Definition 2.6 (Sg). For each set S ⊆ V , we define Sg to be the set of... |

156 | Planetary-scale views on a large instant-messaging network
- Leskovec, Horvitz
- 2008
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Citation Context ...graph. While it may be natural to explore vertices in order of shortest-path distance from the input vertex, such an ordering is a poor choice in graphs of low diameter, such as social network graphs =-=[LH08]-=-. We suggest first processing the vertices that are most likely to occur in short random walks from the input vertex. That is, we consider a vertex to be near the input vertex if it is likely to appea... |

92 | A combinatorial, primal-dual approach to semidefinite programs
- Arora, Kale
- 2007
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Citation Context ... √ logn) through semidefinite programming. Faster algorithms obtaining similar guarantees have been constructed by Arora, Hazan, and Kale [AHK04], Khandekar, Rao, and Vazirani [KRV06], Arora and Kale =-=[AK07]-=-, Orecchia et al. [OSVV08], and Sherman [She09]. 2.1. The algorithm Nibble. The algorithm Nibble works by approximately computing the distribution of a few steps of the random walk starting at a seed ... |

77 |
and Miklós Simonovits. The mixing rate of markov chains, an isoperimetric inequality, and computing the volume. FOCS
- Lovász
- 1990
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Citation Context ...oning algorithm of Hagen and Kahng [HK92] and that implicit in the work of Cheeger [Che70] (see also [Mih89, ST96]), and the probability-based algorithm implicit in the work of Lovász and Simonovits =-=[LS90]-=-, Nibble extracts a cluster from a vector by the following algorithm: 1. Given an n-dimensional real vector x, with one entry x(u) for each vertex u ∈ {1, . . . , n}, compute the permutation π that or... |

71 |
Nearly-linear time algorithms for preconditioning and solving symmetric, diagonally dominant linear systems
- Spielman, Teng
(Show Context)
Citation Context ...n the first nearly linear time algorithm for graph partitioning that produces a partition of nearly optimal balance among those partitions approximating a target conductance. In the papers [ST11] and =-=[ST08]-=-, we use this graph partitioning algorithm to design nearly ∗Received by the editors December 29, 2008; accepted for publication (in revised form) July 22, 2012; published electronically January 10, 2... |

68 |
Graph partitioning using single commodity flows
- Khandekar, Rao, et al.
- 2006
(Show Context)
Citation Context ...V04] improved this to O( √ logn) through semidefinite programming. Faster algorithms obtaining similar guarantees have been constructed by Arora, Hazan, and Kale [AHK04], Khandekar, Rao, and Vazirani =-=[KRV06]-=-, Arora and Kale [AK07], Orecchia et al. [OSVV08], and Sherman [She09]. 2.1. The algorithm Nibble. The algorithm Nibble works by approximately computing the distribution of a few steps of the random w... |

47 | Conductance and Convergence of Markov Chains-A Combinatorial Treatment of Expanders - Mihail - 1989 |

40 | Finding sparse cuts locally using evolving sets
- Andersen, Peres
- 2009
(Show Context)
Citation Context ...efine nearness, Andersen, Chung, and Lang [ACL06] have produced an improved version of our algorithm Nibble, which they call PageRank-Nibble. Their algorithm is further improved by Andersen and Peres =-=[AP09]-=-, who base their algorithm on the volume-biased evolving set process. Following our work, other local algorithms have been designed by Andersen et. al. [ABC+07] for approximately computing personal Pa... |

38 | Spectral sparsification of graphs
- Spielman, Teng
(Show Context)
Citation Context ...it to design the first nearly linear time algorithm for graph partitioning that produces a partition of nearly optimal balance among those partitions approximating a target conductance. In the papers =-=[ST11]-=- and [ST08], we use this graph partitioning algorithm to design nearly ∗Received by the editors December 29, 2008; accepted for publication (in revised form) July 22, 2012; published electronically Ja... |

34 | On partitioning graphs via single commodity flows
- Orecchia, Schulman, et al.
- 2008
(Show Context)
Citation Context ...inite programming. Faster algorithms obtaining similar guarantees have been constructed by Arora, Hazan, and Kale [AHK04], Khandekar, Rao, and Vazirani [KRV06], Arora and Kale [AK07], Orecchia et al. =-=[OSVV08]-=-, and Sherman [She09]. 2.1. The algorithm Nibble. The algorithm Nibble works by approximately computing the distribution of a few steps of the random walk starting at a seed vertex v. It is implicit i... |

29 | Breaking the multicommodity flow barrier for o( √ log n)- approximations to sparsest cut
- Sherman
- 2009
(Show Context)
Citation Context ...ter algorithms obtaining similar guarantees have been constructed by Arora, Hazan, and Kale [AHK04], Khandekar, Rao, and Vazirani [KRV06], Arora and Kale [AK07], Orecchia et al. [OSVV08], and Sherman =-=[She09]-=-. 2.1. The algorithm Nibble. The algorithm Nibble works by approximately computing the distribution of a few steps of the random walk starting at a seed vertex v. It is implicit in the analysis of the... |

28 | Local computation of pagerank contributions - Andersen, Borgs, et al. |

26 |
On Clusterings
- Kannan, Vempala, et al.
- 2004
(Show Context)
Citation Context ...ly to appear in a short random walk from the input vertex. In section 2, we use a local graph exploration process to find a cluster that is near the input vertex. Following Kannan, Vempala, and Vetta =-=[KVV04]-=-, we say that a set of vertices is a good cluster if it has low conductance, that is, if it has many more internal than external edges. We give an efficient local clustering algorithm, Nibble, whose r... |

25 | Local partitioning for directed graphs using PageRank
- Andersen, Chung, et al.
- 2008
(Show Context)
Citation Context ...cal algorithms have been designed by Andersen et. al. [ABC+07] for approximately computing personal PageRank vectors, by Andersen [And08] for finding dense subgraphs, and by Andersen, Chung, and Lang =-=[ACL07]-=- for partitioning directed graphs. 1.2. Nearly linear time algorithms. Our local clustering algorithm provides a A LOCAL CLUSTERING ALGORITHM FOR MASSIVE GRAPHS 3 powerful tool for designing fast grap... |

23 |
O( √ logn) approximation to sparsest cut in Õ(n2) time
- Arora, Hazan, et al.
(Show Context)
Citation Context ...sest cut. Arora, Rao, and Vazirani [ARV04] improved this to O( √ logn) through semidefinite programming. Faster algorithms obtaining similar guarantees have been constructed by Arora, Hazan, and Kale =-=[AHK04]-=-, Khandekar, Rao, and Vazirani [KRV06], Arora and Kale [AK07], Orecchia et al. [OSVV08], and Sherman [She09]. 2.1. The algorithm Nibble. The algorithm Nibble works by approximately computing the distr... |

19 | On the NP-Completeness of Some Graph Cluster Measures
- Śıma, Schaeffer
- 2006
(Show Context)
Citation Context ...mization problem: Given an undirected graph G and a parameter φ, find a cluster C such that Φ (C) ≤ φ or determine that no such cluster exists. The problem is NP-complete (see, for example, [LR99] or =-=[SS06]-=-). But, approximation algorithms exist. Leighton and Rao [LR99] used linear programming to obtainO(log n)-approximations of the sparsest cut. Arora, Rao, and Vazirani [ARV04] improved this to O( √ log... |

14 | A local algorithm for finding dense subgraphs
- Andersen
- 2010
(Show Context)
Citation Context ...n the volume-biased evolving set process. Following our work, other local algorithms have been designed by Andersen et. al. [ABC+07] for approximately computing personal PageRank vectors, by Andersen =-=[And08]-=- for finding dense subgraphs, and by Andersen, Chung, and Lang [ACL07] for partitioning directed graphs. 1.2. Nearly linear time algorithms. Our local clustering algorithm provides a A LOCAL CLUSTERIN... |

9 |
Finding local communities in protein networks
- Voevodski, Teng, et al.
(Show Context)
Citation Context ..., and Lang [ACL06] and Andersen and Peres [AP09], may not be immediately useful in practice, we believe practical algorithms may be derived from them by making less conservative choices of parameters =-=[VTX09]-=-. Our local clustering algorithm provides an exciting new primitive for developing nearly linear time graph algorithms. Because its running time is proportional to the size of the cluster it produces ... |

8 | A Metascalable Computing Framework for Large Spatiotemporal-scale Atomistic Simulations - Nomura, Seymour, et al. - 2009 |

5 | Towards an sdp-based approach to spectral methods: A nearly-linear-time algorithm for graph partitioning and decomposition
- Orecchia, Vishnoi
- 2011
(Show Context)
Citation Context ...ime partitioning algorithm that finds an approximate sparsest cut with approximately optimal balance. A faster algorithm with better output guarantees has since been developed by Orecchia and Vishnoi =-=[OV11]-=-. In our first companion paper [ST11], we apply this new partitioning algorithm to develop a nearly linear time algorithm for producing spectral sparsifiers of graphs. We begin that paper by extending... |

1 | Transistor Count, http://en.wikipedia.org/wiki/Transistor count - Wikipedia - 2010 |