## Connected Components in Random Graphs with Given Expected Degree Sequences

Venue: | ANNALS OF COMBINATORICS |

Citations: | 139 - 15 self |

### BibTeX

@ARTICLE{Chung_connectedcomponents,

author = {Fan Chung and Linyuan Lu},

title = { Connected Components in Random Graphs with Given Expected Degree Sequences},

journal = {ANNALS OF COMBINATORICS},

year = {},

pages = {125--145}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

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Citation Context ...o illustrate that both bounds are best possible. 1 Introduction The primary subject in the study of random graph theory is the classical random graph G(n, p), as introduced by Erdős and Rényi in 1959 =-=[19]-=-. In G(n, p), every pair of a set of n vertices is chosen to be an edge with probability p. Such random graphs are fundamental and useful for modeling problems in many applications. However, a random ... |

1857 | Random Graphs
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Citation Context ...ollowing model, so called the configuration model, is often used to construct a random graph with a prescribed degree sequence. It was first introduced by Bender and Canfield [9], refined by Bollobás =-=[10]-=- and also Wormald [35]. A random graph G with given degrees dv is formed by first associating to each vertex v a set Sv of dv nodes, then considering the disjoint union N of Sv and taking a random mat... |

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Citation Context ...nce of a connected spanning graph on S is at most ∑ Pr(T) = ∑ T T ∏ (vi j vi l )∈E(T) where T ranges over all spanning trees on S. wijwil ρ, By a generalized version of celebrated matrix-tree Theorem =-=[34]-=-, the above sum equals the determinant of any k − 1 by k − 1 principal sub-matrix of the matrix D − A, where A is the weight matrix 11⎛ A = ⎜ ⎝ 0 wi1wi2ρ · · · wi1wik ρ wi2wi1ρ 0 · · · wi2wik ρ . . .... |

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Citation Context ...g t = λ ν . This completes the proof of Lemma 1. We note that the special case of ai = 1 for all i is the usual inequality that is included in most books in random graph theory and probability (e.g., =-=[24]-=-). As immediate consequences of Lemma 1, the following facts then follow. Fact 1: For a graph G in G(w), with probability 1 − e −c2 /2 , the number di of edges incident to a vertex vi satisfies □ di >... |

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Citation Context ...h supported in part by NSF Grant DMS 0100472 1It has been observed that many real graphs occurring in the Internet, social sciences, computational biology and nature have degrees obeying a power law =-=[1, 2, 3, 7, 8, 12, 13, 20, 21, 25, 26, 36]-=-. Namely, the fraction of vertices with degree d is proportional to 1/d α for some constant α > 0. Although here we consider random graphs with general expected degree distributions, special emphasis ... |

350 | A random graph model for massive graphs
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Citation Context ...h supported in part by NSF Grant DMS 0100472 1It has been observed that many real graphs occurring in the Internet, social sciences, computational biology and nature have degrees obeying a power law =-=[1, 2, 3, 7, 8, 12, 13, 20, 21, 25, 26, 36]-=-. Namely, the fraction of vertices with degree d is proportional to 1/d α for some constant α > 0. Although here we consider random graphs with general expected degree distributions, special emphasis ... |

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Citation Context ...h supported in part by NSF Grant DMS 0100472 1It has been observed that many real graphs occurring in the Internet, social sciences, computational biology and nature have degrees obeying a power law =-=[1, 2, 3, 7, 8, 12, 13, 20, 21, 25, 26, 36]-=-. Namely, the fraction of vertices with degree d is proportional to 1/d α for some constant α > 0. Although here we consider random graphs with general expected degree distributions, special emphasis ... |

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Citation Context ... just a random graph with expected degree sequence (pn, pn, . . .,pn). The random graph G(w) is different from the random graphs with a prescribed degree sequence as considered by Molloy and Reed. In =-=[31, 32]-=-, Molloy and Reed obtained results on the sizes of connected components for random graphs with prescribed degree sequences which satisfy certain “smoothing” 2conditions. There are also a number of ev... |

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The asymptotic number of labeled graphs with given degree sequences
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Citation Context ... In the literature, the following model, so called the configuration model, is often used to construct a random graph with a prescribed degree sequence. It was first introduced by Bender and Canfield =-=[9]-=-, refined by Bollobás [10] and also Wormald [35]. A random graph G with given degrees dv is formed by first associating to each vertex v a set Sv of dv nodes, then considering the disjoint union N of ... |

137 | The size of the giant component of a random graph with a given degree sequence, Combinatorics, Probabililty and Computing 7
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Citation Context ... just a random graph with expected degree sequence (pn, pn, . . .,pn). The random graph G(w) is different from the random graphs with a prescribed degree sequence as considered by Molloy and Reed. In =-=[31, 32]-=-, Molloy and Reed obtained results on the sizes of connected components for random graphs with prescribed degree sequences which satisfy certain “smoothing” 2conditions. There are also a number of ev... |

109 | Extracting large-scale knowledge bases from the Web
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Citation Context ...of complete bipartite subgraphs, as has been observed in the WWW graph, whereas several other models do not. This (and the linear growth variants model) has the similar drawback as the first model in =-=[27]-=-. The out-degree of every vertex is always a constant. Edges and vertices in the exponential growth copying model increase exponentially. Aiello et al. described a general random graph evolution proce... |

85 | A general model of Web graphs
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Citation Context ...ences which satisfy certain “smoothing” 2conditions. There are also a number of evolution models for generating a powerlaw degree random graphs as in Bollobás, Spencer et al. [11], Cooper and Freeze =-=[17]-=- and Aiello, Chung and Lu [2]. In Section 8, we will describe and compare these models and related results. Here we give some definitions. The expected average degree d of a random graph G in G(w) is ... |

47 | Eigenvalues of random power law graphs - Chung, Lu, et al. |

45 | The diameter of random sparse graphs
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Citation Context ...ing with random graphs with given expected degree distribution are useful not only for modeling and analyzing realistic graphs but also leading to improvements for problems on classical random graphs =-=[14, 29]-=-. In this paper, we consider the following class of random graphs. We start with a given degree sequence w = (w1, w2, . . . , wn). The vertex vi is assigned vertex weight wi. The edges are chosen inde... |

37 | The Diameter of Random Massive Graphs
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- 2000
(Show Context)
Citation Context ...ing with random graphs with given expected degree distribution are useful not only for modeling and analyzing realistic graphs but also leading to improvements for problems on classical random graphs =-=[14, 29]-=-. In this paper, we consider the following class of random graphs. We start with a given degree sequence w = (w1, w2, . . . , wn). The vertex vi is assigned vertex weight wi. The edges are chosen inde... |

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Citation Context |

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21 |
Gráfok elő´irt fokú pontokkal (Graphs with points of prescribed degrees
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Citation Context ... wk . (1) < ∑ k wk so that pij ≤ 1 for < ∑ k wk implies that the sequence wi is graphic (in the sense that it satisfies the necessary and sufficient condition for a sequence to be realized by a graph =-=[18]-=-) except that we do not require the wi’s to be integers. We denote a random graph with a given expected degree sequence w by G(w). For example, the typical random graph G(n, p) (see [19]) on n vertice... |

12 |
The Erdős Number Project, http://www.oakland.edu/grossman/erdoshp.html
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Citation Context ...ee k proportional to kα . For example, the collaboration graph consists of 337,000 authors in Mathematics Review as vertices and collaborations as edges, as described in the webpage of Jerry Grossman =-=[22]-=- at http://www.oakland.edu/∼grossman/trivia.html. From Figure 2, we can see that the degree sequence of the collaboration graph can be approximated by a power law with power 2.2. 100000 "collab1.degre... |

11 |
The Degree Sequence of a Scale-free Random
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Citation Context ...h prescribed degree sequences which satisfy certain “smoothing” 2conditions. There are also a number of evolution models for generating a powerlaw degree random graphs as in Bollobás, Spencer et al. =-=[11]-=-, Cooper and Freeze [17] and Aiello, Chung and Lu [2]. In Section 8, we will describe and compare these models and related results. Here we give some definitions. The expected average degree d of a ra... |

6 | A power law for cells - Azevedo, Leroi - 2001 |

5 | The Structure of Scientific Collaboration Networks - J |

4 |
Random evolution in massive graphs, Extended abstract appeared in: The 42th Annual Symposium on Foundation
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Citation Context |

3 |
Stochastic models for the Web graph, to appear
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Citation Context ...e degree distribution with a power of 3, regardless of m. A power law with power 3 for the degree distribution of this model was independently derived and proved by Ballobás et al. [11]. Kumar at el. =-=[28]-=- proposed three evolution models — “linear growth copying”, “exponential growth copying”, and “linear growth variants”. The Linear growth coping model adds one new vertex with d out-links at a time. T... |

2 | Small world graphs and random graphs, preprint - Chung, Lu |

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1 |
Concentration. Probabilistic methods for algorithmic discrete mathematics
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Citation Context ..., an}. Pr(X < E(X) − λ) ≤ e −λ2 /2ν (2) Pr(X > E(X) + λ) ≤ e − λ 2 2(ν+aλ/3) (3) Inequality (3) is a corollary of a general concentration inequality ( see Theorem 2.7 in the survey paper by McDiarmid =-=[30]-=-). Inequality (2) which is a slight improvement of the inequality in [30] can be proved as follows. Proof: For any 0 ≤ p ≤ 1, and x ≥ 0, we denote f(x) = px+ln(1−p+pe −x ) and g(x) = px 2 /2. Then we ... |

1 |
Some problems in the enumberation of labelled graphs, Doctoral thesis
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Citation Context ...led the configuration model, is often used to construct a random graph with a prescribed degree sequence. It was first introduced by Bender and Canfield [9], refined by Bollobás [10] and also Wormald =-=[35]-=-. A random graph G with given degrees dv is formed by first associating to each vertex v a set Sv of dv nodes, then considering the disjoint union N of Sv and taking a random matching M on N. The numb... |