A Local Clustering Algorithm for Massive Graphs and Its Application to Nearly Linear Time Graph Partitioning

@article{Spielman2013ALC,
  title={A Local Clustering Algorithm for Massive Graphs and Its Application to Nearly Linear Time Graph Partitioning},
  author={Daniel A. Spielman and Shang-Hua Teng},
  journal={SIAM J. Comput.},
  year={2013},
  volume={42},
  pages={1-26}
}
We study the design of local algorithms for massive graphs. A local graph algorithm is one that finds a solution containing or near a given vertex without looking at the whole graph. We present a local clustering algorithm. Our algorithm finds a good cluster---a subset of vertices whose internal connections are significantly richer than its external connections---near a given vertex. The running time of our algorithm, when it finds a nonempty local cluster, is nearly linear in the size of the… 
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References

SHOWING 1-10 OF 40 REFERENCES
Finding sparse cuts locally using evolving sets
TLDR
A randomized local partitioning algorithm is introduced that finds a sparse cut by simulating the volume-biased evolving set process, which is a Markov chain on sets of vertices and the expected value of the work/volume ratio is polylognoparen(φ-1/2).
Local Graph Partitioning using PageRank Vectors
TLDR
An improved algorithm for computing approximate PageRank vectors, which allows us to find a cut with conductance at most oslash and approximately optimal balance in time O(m log4 m/oslash) in time proportional to its size.
Local Partitioning for Directed Graphs Using PageRank
TLDR
It is proved that by computing a personalized PageRank vector in a directed graph, starting from a single seed vertex within a set S that has conductance at most α, and by performing a sweep over that vector, one can obtain a set of vertices S′ with conductance.
A local algorithm for finding dense subgraphs
TLDR
A local algorithm for finding subgraphs with high density, according to a measure of density introduced by Kannan and Vinay [1999], which is local in the sense that it adaptively explores a region of the graph near the starting vertex.
On the NP-Completeness of Some Graph Cluster Measures
TLDR
It is proved that the decision problems associated with the optimization tasks of finding clusters that are optimal with respect to these fitness measures are NP-complete.
Spectral Sparsification of Graphs
TLDR
It is proved that every graph has a spectral sparsifier of nearly linear size, and an algorithm is presented that produces spectralSparsifiers in time $O(m\log^{c}m)$, where $m$ is the number of edges in the original graph and $c$ is some absolute constant.
Finding local communities in protein networks
TLDR
A tool, named Local Protein Community Finder, which quickly finds a community close to a queried protein in any network available from BioGRID or specified by the user, using two new local clustering algorithms Nibble and PageRank-Nibble, which look for a good cluster among the most popular destinations of a short random walk from the queried vertex.
Spectral partitioning works: planar graphs and finite element meshes
  • D. Spielman, S. Teng
  • Computer Science, Mathematics
    Proceedings of 37th Conference on Foundations of Computer Science
  • 1996
TLDR
It is proved that spectral partitioning techniques can be used to produce separators whose ratio of vertices removed to edges cut is O(/spl radic/n) for bounded-degree planar graphs and two-dimensional meshes and O(n/sup 1/d/) for well-shaped d-dimensional mesh.
Breaking the Multicommodity Flow Barrier for O(vlog n)-Approximations to Sparsest Cut
  • Jonah Sherman
  • Computer Science
    2009 50th Annual IEEE Symposium on Foundations of Computer Science
  • 2009
TLDR
The core of the algorithm is a stronger, algorithmic version of Arora et al.'s structure theorem, where it is shown that matching-chaining argument at the heart of their proof can be viewed as an algorithm that finds good augmenting paths in certain geometric multicommodity flow networks.
Random Walks in a Convex Body and an Improved Volume Algorithm
TLDR
A randomized algorithm using O(n7 log’ n) separation calls to approximate the volume of a convex body with a fixed relative error is given and the mixing rate of Markov chains from finite to arbitrary Markov Chains is analyzed.
...
...