## A General Model of Web Graphs (2003)

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Citations: | 89 - 7 self |

### BibTeX

@MISC{Cooper03ageneral,

author = {Colin Cooper and Alan Frieze},

title = {A General Model of Web Graphs},

year = {2003}

}

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### Abstract

We describe a very general model of a random graph process whose proportional degree sequence obeys a power law. Such laws have recently been observed in graphs associated with the world wide web.

### Citations

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Citation Context ...a)tl _> t /2 logt) = O(t -) (4) for any constant K > 0. Similarly, r/(t) has expectation Ot and is the sum of t independent random variables, each bounded by max{j0, j}. Hence, by Hoeffding's theorem =-=[12]-=-, rr(Iv(t ) - Ot I > t /2 logt) = O(t -:) (5) for any constant K > 0. We remind the reader that D(t) is the number of vertices of degree k at step t and that D(t) is its expectation. Here Do(t) = 0 fo... |

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Citation Context ... (ii) Alpha-Beta graphs: Standard random graph models with atypical degree sequences. See eg: [2], [8], [17], [18]. (iii) Lattice graphs: Graphs generated by perturbing regular lattices. See eg: [4], =-=[13]-=-. We describe the evolution of a random (multi-)graph G(t) which is an example of the type of model referred to as a web graph. Initially, at step t = 0, there is a single vertex v0. At any step t = 1... |

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Citation Context ...e addition of new vertices and/or edges at each step t. See eg: [1], [3], [5], [6], [7], [10], [15], [16]. (ii) Alpha-Beta graphs: Standard random graph models with atypical degree sequences. See eg: =-=[2]-=-, [8], [17], [18]. (iii) Lattice graphs: Graphs generated by perturbing regular lattices. See eg: [4], [13]. We describe the evolution of a random (multi-)graph G(t) which is an example of the type of... |

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Citation Context ... of new vertices and/or edges at each step t. See eg: [1], [3], [5], [6], [7], [10], [15], [16]. (ii) Alpha-Beta graphs: Standard random graph models with atypical degree sequences. See eg: [2], [8], =-=[17]-=-, [18]. (iii) Lattice graphs: Graphs generated by perturbing regular lattices. See eg: [4], [13]. We describe the evolution of a random (multi-)graph G(t) which is an example of the type of model refe... |

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Citation Context ...w vertices and/or edges at each step t. See eg: [1], [3], [5], [6], [7], [10], [15], [16]. (ii) Alpha-Beta graphs: Standard random graph models with atypical degree sequences. See eg: [2], [8], [17], =-=[18]-=-. (iii) Lattice graphs: Graphs generated by perturbing regular lattices. See eg: [4], [13]. We describe the evolution of a random (multi-)graph G(t) which is an example of the type of model referred t... |

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Citation Context ...ce equation (2) Re-writing (2) we see that for k _ j0, d satisfies d = d_ c + d(k - 1) J e + f(k - j) (13) a + bk +sd_jqj a + bk ' j=l which is a linear difference equation with rational coefficients =-=[19]-=-. The general solution for dk is a power law, i.e. there are constants xL,xu such that Ak-xL _ _ Lemma 2. In the cases where j = i (a new vertex generates a single edge) or f = 0 (old initial vertices... |

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Citation Context ...describe small world phenomena; see for example [11] for an introduction to this topic. These axe: (i) Web graphs: Graphs evolving by the addition of new vertices and/or edges at each step t. See eg: =-=[1]-=-, [3], [5], [6], [7], [10], [15], [16]. (ii) Alpha-Beta graphs: Standard random graph models with atypical degree sequences. See eg: [2], [8], [17], [18]. (iii) Lattice graphs: Graphs generated by per... |

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Citation Context ...[16]. (ii) Alpha-Beta graphs: Standard random graph models with atypical degree sequences. See eg: [2], [8], [17], [18]. (iii) Lattice graphs: Graphs generated by perturbing regular lattices. See eg: =-=[4]-=-, [13]. We describe the evolution of a random (multi-)graph G(t) which is an example of the type of model referred to as a web graph. Initially, at step t = 0, there is a single vertex v0. At any step... |

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Citation Context ...omena; see for example [11] for an introduction to this topic. These axe: (i) Web graphs: Graphs evolving by the addition of new vertices and/or edges at each step t. See eg: [1], [3], [5], [6], [7], =-=[10]-=-, [15], [16]. (ii) Alpha-Beta graphs: Standard random graph models with atypical degree sequences. See eg: [2], [8], [17], [18]. (iii) Lattice graphs: Graphs generated by perturbing regular lattices. ... |

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Citation Context ...ibe small world phenomena; see for example [11] for an introduction to this topic. These axe: (i) Web graphs: Graphs evolving by the addition of new vertices and/or edges at each step t. See eg: [1], =-=[3]-=-, [5], [6], [7], [10], [15], [16]. (ii) Alpha-Beta graphs: Standard random graph models with atypical degree sequences. See eg: [2], [8], [17], [18]. (iii) Lattice graphs: Graphs generated by perturbi... |

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Citation Context ...ition of new vertices and/or edges at each step t. See eg: [1], [3], [5], [6], [7], [10], [15], [16]. (ii) Alpha-Beta graphs: Standard random graph models with atypical degree sequences. See eg: [2], =-=[8]-=-, [17], [18]. (iii) Lattice graphs: Graphs generated by perturbing regular lattices. See eg: [4], [13]. We describe the evolution of a random (multi-)graph G(t) which is an example of the type of mode... |

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Citation Context ...cently been observed in graphs associated with the world wide web. I Introduction Essentially, there are tkree types of random graph or digraph used to describe small world phenomena; see for example =-=[11]-=- for an introduction to this topic. These axe: (i) Web graphs: Graphs evolving by the addition of new vertices and/or edges at each step t. See eg: [1], [3], [5], [6], [7], [10], [15], [16]. (ii) Alph... |

4 | The web as a graph. www.almaden.ibm.com - Kumar, Raghavan, et al. |

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2 |
The degree sequence of a scale free random graph process
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(Show Context)
Citation Context ...mall world phenomena; see for example [11] for an introduction to this topic. These axe: (i) Web graphs: Graphs evolving by the addition of new vertices and/or edges at each step t. See eg: [1], [3], =-=[5]-=-, [6], [7], [10], [15], [16]. (ii) Alpha-Beta graphs: Standard random graph models with atypical degree sequences. See eg: [2], [8], [17], [18]. (iii) Lattice graphs: Graphs generated by perturbing re... |

2 |
The diameter of a scale free random graph
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(Show Context)
Citation Context ...world phenomena; see for example [11] for an introduction to this topic. These axe: (i) Web graphs: Graphs evolving by the addition of new vertices and/or edges at each step t. See eg: [1], [3], [5], =-=[6]-=-, [7], [10], [15], [16]. (ii) Alpha-Beta graphs: Standard random graph models with atypical degree sequences. See eg: [2], [8], [17], [18]. (iii) Lattice graphs: Graphs generated by perturbing regular... |

1 |
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Citation Context ...a mixture of in-degree and out-degree, and the estimation of ci,j, the expected proportion of vertices of in-degree i and out-degree j is not considered here, but is the subject of a subsequent paper =-=[9]-=-. The parameters of the undirected model Our undirected model G(t) has sampling parameters et,/, if, 5, p, q whose meaning is given below: Choice of procedure at step t. et: Probability that an OLD no... |

1 | T.Luczak and A.Ruciiski, Random Graphs - Janson - 2000 |

1 |
The web as a graph. rw. almadan. ibm. corn
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(Show Context)
Citation Context ... see for example [11] for an introduction to this topic. These axe: (i) Web graphs: Graphs evolving by the addition of new vertices and/or edges at each step t. See eg: [1], [3], [5], [6], [7], [10], =-=[15]-=-, [16]. (ii) Alpha-Beta graphs: Standard random graph models with atypical degree sequences. See eg: [2], [8], [17], [18]. (iii) Lattice graphs: Graphs generated by perturbing regular lattices. See eg... |

1 |
Stochastic models for the web graph. ww. a/maden. ibm. corn
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(Show Context)
Citation Context ...or example [11] for an introduction to this topic. These axe: (i) Web graphs: Graphs evolving by the addition of new vertices and/or edges at each step t. See eg: [1], [3], [5], [6], [7], [10], [15], =-=[16]-=-. (ii) Alpha-Beta graphs: Standard random graph models with atypical degree sequences. See eg: [2], [8], [17], [18]. (iii) Lattice graphs: Graphs generated by perturbing regular lattices. See eg: [4],... |

1 |
Problems and Theorems in Analysis I
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Citation Context ...or {qj} is not periodic, then the solutions of qS(z) -- 0 are i) An un-repeated root at z = 1, ii) l - 1 other (possibly repeated) roots A satisfying IlsProof We note the following (see P61ya & Szeg6 =-=[20]-=- p106 16,17). A polynomial f(z) of the form f(z) ---- Z n -- plZ n-1 -- p2z n-2 ..... Pn-lZ -- Pn, 12 where Pi _ 0, i -- 1, ..., n and p + ... + Pn > 0 has just one positive zero 4. All other zeroes z... |

1 | Distribution of vertex degree - Cooper - 2002 |