## Finding good nearly balanced cuts in power law graphs (2004)

Citations: | 14 - 2 self |

### BibTeX

@TECHREPORT{Lang04findinggood,

author = {Kevin Lang},

title = {Finding good nearly balanced cuts in power law graphs},

institution = {},

year = {2004}

}

### Years of Citing Articles

### OpenURL

### Abstract

In power law graphs, cut quality varies inversely with cut balance. Using some million node social graphs as a test bed, we empirically investigate this property and its implications for graph partitioning. We use six algorithms, including Metis and MQI (state of the art methods for finding bisections and quotient cuts) and four relaxation/rounding methods. We find that an SDP relaxation avoids the Spectral method’s tendency to break off tiny pieces of the graph. We also find that a flow-based rounding method works better than hyperplane rounding. 1

### Citations

1383 | Fast approximate energy minimization via graph cuts
- Boykov, Veksler, et al.
- 2001
(Show Context)
Citation Context ...cut that we saw in any iteration. Otherwise, go to back step 1 and try a new direction. This flow problem construction seems pretty obvious and is probably folklore; something like it is described in =-=[2]-=-. We solve the flow problem usinghi pr, Cherkassky and Goldberg’s implementation of the highest-node variant of the push-relabel algorithm [5]. Its worst case run time is O(n 2.5 ), but it seems to ru... |

189 | The average distance in a random graph with given expected degrees
- Chung, Lu
(Show Context)
Citation Context ...phs there are good cuts with bad balance and bad cuts with good balance, but there are no good cuts with good balance. 1 Graph theorists have noticed that power law graphs have this kind of structure =-=[6]-=-, but the problem has not been discussed much in the empirical literature, which has mostly focused on finite element meshes and other classes of graphs where the best quotient cut does not necessaril... |

172 |
λ1, isoperimetric inequalities for graphs, and superconcentrators
- Alon, Milman
- 1985
(Show Context)
Citation Context ... corresponding to the second smallest eigenvalue of the graph’s Laplacian matrix [7]. Hence solving SDP-1 is generally known as the spectral method. In the mid 1980’s it was proved by Alon and Milman =-=[1]-=- and other that the spectral method yields solutions that are at worst quadratically bad (see also [16]). Later analysis by Spielman and Teng showed why the spectral method works especially well on pl... |

150 | On implementing push-relabel method for the maximum flow problem
- Cherkassy, Goldberg
- 1995
(Show Context)
Citation Context ...d is probably folklore; something like it is described in [2]. We solve the flow problem usinghi pr, Cherkassky and Goldberg’s implementation of the highest-node variant of the push-relabel algorithm =-=[5]-=-. Its worst case run time is O(n 2.5 ), but it seems to run in nearly linear time in practice, with a very good constant factor. 5srun time (seconds) 100000 10000 1000 100 10 1 0.1 Solving SDP-2 with ... |

143 |
Lower bounds for the partitioning of graphs
- Donath, Hoffman
- 1973
(Show Context)
Citation Context ...average squared magnitude is 1.0. It has been known for at least 30 years that the solution to SDP-1 is the eigenvector corresponding to the second smallest eigenvalue of the graph’s Laplacian matrix =-=[7]-=-. Hence solving SDP-1 is generally known as the spectral method. In the mid 1980’s it was proved by Alon and Milman [1] and other that the spectral method yields solutions that are at worst quadratica... |

103 | A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization
- Burer, Monteiro
(Show Context)
Citation Context ...them using one of many SDP solvers, notably Helmberg and Rendl’s program SBmethod [12], and Burer and Monteiro’s special-purpose solver forSDP-2 which can handle graphs with more than a million nodes =-=[4]-=-. 2 We frequently hear the suggestion that Malik and Shi’s “normalized cut” version of the spectral method would solve this problem of unbalanced solutions. It does not, and in fact, figure 1-B was dr... |

54 | Throughput and congestion in power-law graphs
- Gkantsidis, Mihail, et al.
(Show Context)
Citation Context ...rge pieces would require cutting through the high-expansion core, and the many edges that would necessarily be cut make it impossible for the quotient cut score to be very good in that case (see also =-=[8]-=-). An inverse tradeoff between cut quality and balance appears clearly in figure 2, which contains a scatter plot of cuts in a 1.9 million node subgraph of the Yahoo IM graph. The y-axis shows the qua... |

17 | Local minima and convergence in low-rank semidefinite programming
- Burer, Monteiro
- 2005
(Show Context)
Citation Context ...for 5 dimensions, then 4 dimensions apparently are not enough. On some standard mesh-like benchmark graphs, this approach seems to recover the true dimensionality. Furthermore, on several graphs 3 In =-=[3]-=- Burer and Monteiro discuss this issue in detail and also explain why their approach does not get stuck in local minima. 4 Specifically, we used the special purpose program minbis which is part of SDP... |