## Eigenvalues of Random Power Law Graphs (2003)

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Citations: | 47 - 7 self |

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@MISC{Chung03eigenvaluesof,

author = {Fan Chung and Linyuan Lu and Van Vu},

title = {Eigenvalues of Random Power Law Graphs },

year = {2003}

}

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### Abstract

Many graphs arising in various information networks exhibit the “power law” behavior – the number of vertices of degree k is proportional to k −β for some positive β. We show that if β > 2.5, the largest eigenvalue of a random power law graph is almost surely (1+o(1)) √ m where m is the maximum degree. When 2 < β < 2.5, the largest eigenvalue is heavily concentrated at cm 3−β for some constant c depending on β and the average degree. This result follows from a more general theorem which shows that the largest eigenvalue of a random graph with a given expected degree sequence is determined by m, the maximum degree, and ˜ d, the weighted average of the squares of the expected degrees. We show that λ is almost surely (1 + o(1)) max { ˜ d, √ m} provided some minor condition is satisfied. Our results have implications on the usage of spectral techniques in many areas related to pattern detection and information retrieval.

### Citations

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Citation Context ...l methods are central in detecting clusters and finding patterns in various applications. The eigenvalues of the adjacency matrices of various realistic power law graphs were computed and examined in =-=[8, 9, 11]-=-. Faloutsos et al. [8] conjectured a power law distribution for eigenvalues of power law graphs. For a fixed value β > 1, we say that a graph is a power law graph with exponent β if the number of vert... |

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Citation Context .... Lu, and V. Vu degree k is proportional to k−β . We note that for most realistic graphs, their power law models usually have exponents β falling between 2 and 3. For example, various Internet graphs =-=[14]-=- have exponents between 2.1and2.4. The Hollywood graph [4] has exponent β ∼ 2.3. The telephone call graph [1] has exponent β = 2.1. Recently, Mihail and Papadimitriou [16] showed that the largest eige... |

166 |
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147 | The large-scale organization of metabolic networks, Nature 407 (6804 - Jeong, Tombor, et al. - 2000 |

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Citation Context ...at β = 2.5. This result for power law graphs is an immediate consequence of a general result for eigenvalues of random graphs with arbitrary degree distribution. We will use a random graph model from =-=[5]-=-, which is a generalization of the ErdősRényi model, for random graphs with given expected degrees w1, w2,...,wn. The largest eigenvalue λ1 of the adjacency matrix of a random graph in this model depe... |

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Citation Context ...1) For β ≥ 3, suppose the maximum degree m satisfies m > d 2 log 3 n, (5.1) where d is the average degree. Then almost surely the largest eigenvalue of the random power law graph G is (1 + o(1)) √ m. =-=(2)-=- For 3 > β > 2.5, suppose m satisfies m > d β−2 β−2.5 log 3 β−2.5 n. (5.2) Then almost surely the largest eigenvalue of the random power law graph G is (1 + o(1)) √ m. (3) For 2 < β < 2.5 and m > log ... |

53 |
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Citation Context ...w graph G is (1 + o(1)) √ m. (2) For 3 > β > 2.5, suppose m satisfies m > d β−2 β−2.5 log 3 β−2.5 n. (5.2) Then almost surely the largest eigenvalue of the random power law graph G is (1 + o(1)) √ m. =-=(3)-=- For 2 < β < 2.5 and m > log 3 2.5−β n, almost surely the largest eigenvalue is (1 + o(1)) d. ˜ d (4) For k < n( mlogn )β−1 and β > 2.5, almost surely the k largest eigenvalues of the random power law... |

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Citation Context ...l methods are central in detecting clusters and finding patterns in various applications. The eigenvalues of the adjacency matrices of various realistic power law graphs were computed and examined in =-=[8, 9, 11]-=-. Faloutsos et al. [8] conjectured a power law distribution for eigenvalues of power law graphs. For a fixed value β > 1, we say that a graph is a power law graph with exponent β if the number of vert... |

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- 2001
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Citation Context ...graphs arising in numerous arenas. Graphs with power law degree distribution are ubiquitous as observed in the Internet, the telecommunications graphs, email graphs and in various biological networks =-=[2, 3, 4, 8, 12, 13, 14]-=-. One of the basic problems concerns the distribution of the eigenvalues of power law graphs. In addition to theoretical interest, spectral methods are central in detecting clusters and finding patter... |

30 |
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- Goh, Kahng, et al.
- 2001
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Citation Context ...l methods are central in detecting clusters and finding patterns in various applications. The eigenvalues of the adjacency matrices of various realistic power law graphs were computed and examined in =-=[8, 9, 11]-=-. Faloutsos et al. [8] conjectured a power law distribution for eigenvalues of power law graphs. For a fixed value β > 1, we say that a graph is a power law graph with exponent β if the number of vert... |

19 |
Random evolution in massive graphs, Handbook of Massive Data Sets, Volume 2, (Eds
- Aiello, Chung, et al.
- 2001
(Show Context)
Citation Context ...graphs arising in numerous arenas. Graphs with power law degree distribution are ubiquitous as observed in the Internet, the telecommunications graphs, email graphs and in various biological networks =-=[2, 3, 4, 8, 12, 13, 14]-=-. One of the basic problems concerns the distribution of the eigenvalues of power law graphs. In addition to theoretical interest, spectral methods are central in detecting clusters and finding patter... |

15 |
Gráfok elő´irt fokú pontokkal (Graphs with points of prescribed degrees
- Erdős, Gallai
- 1961
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Theorie der algebraischen Gleichungen, II (zweite Auflage), de Gruyter
- Perron
- 1933
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On the eigenvalue power law, preprint
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Citation Context ...ple, various Internet graphs [14] have exponents between 2.1and2.4. The Hollywood graph [4] has exponent β ∼ 2.3. The telephone call graph [1] has exponent β = 2.1. Recently, Mihail and Papadimitriou =-=[16]-=- showed that the largest eigenvalue of a power law graph with exponent β has power law distribution if the exponent β of the power law graph satisfies β > 3. In this paper, we will show that the large... |