## The Web as a graph: measurements, models, and methods (1999)

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Citations: | 312 - 11 self |

### BibTeX

@MISC{Kleinberg99theweb,

author = {Jon M. Kleinberg and Ravi Kumar and Prabhakar Raghavan and Sridhar Rajagopalan and Andrew S. Tomkins},

title = {The Web as a graph: measurements, models, and methods},

year = {1999}

}

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### Abstract

. The pages and hyperlinks of the World-Wide Web may be viewed as nodes and edges in a directed graph. This graph is a fascinating object of study: it has several hundred million nodes today, over a billion links, and appears to grow exponentially with time. There are many reasons --- mathematical, sociological, and commercial --- for studying the evolution of this graph. In this paper we begin by describing two algorithms that operate on the Web graph, addressing problems from Web search and automatic community discovery. We then report a number of measurements and properties of this graph that manifested themselves as we ran these algorithms on the Web. Finally, we observe that traditional random graph models do not explain these observations, and we propose a new family of random graph models. These models point to a rich new sub-field of the study of random graphs, and raise questions about the analysis of graph algorithms on the Web. 1 Overview Few events in the history of comput...

### Citations

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Citation Context ...dges) to other pages, making for a total of several billion hyperlinks in all.sThere are several reasons for studying the Web graph. The structure of this graph has already led to improved Web search =-=[6,8, 21, 29]-=-, more accurate topic-classi cation algorithms [11] and has inspired algorithms for enumerating emergent cyber-communities [23]. The hyperlinks themselves represent afecund source of sociological info... |

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Citation Context ...dges) to other pages, making for a total of several billion hyperlinks in all.sThere are several reasons for studying the Web graph. The structure of this graph has already led to improved Web search =-=[6,8, 21, 29]-=-, more accurate topic-classi cation algorithms [11] and has inspired algorithms for enumerating emergent cyber-communities [23]. The hyperlinks themselves represent afecund source of sociological info... |

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Citation Context ... two examples of projects in this space. Some other examples are W3QS [22], WebQuery [8], Weblog [24], and ParaSite/Squeal [29]. Traditional data mining research (see for instance Agrawal and Srikant =-=[2]-=-) focuses largely on algorithms for nding association rules and related statistical correlation measures in a given dataset. However, e cient methods such as a priori [2] or even more general methodol... |

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Citation Context ... Zip an (inverse polynomial) distributions [12, 17, 26, 31]. This and other measurements of the frequency of occurrence of certain structures suggest that traditional random graph models such asG n�p =-=[7]-=- are likely to do a poor job of modeling the Web graph. In Section 4 we laydown a framework for a class of random graph models, and give evidence that at least some of our observations about the Web (... |

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Citation Context ...wn independently. We conclude in Section 5 with a number of directions for further work. 1.2 Related work Analysis of the structure of the Web graph has been used to enhance the quality of Web search =-=[5, 6, 8, 9, 21, 29]-=-. The topics of pages pointed to by aWeb page can be used to improve the accuracy of determining the (unknown) topic of this page in the setting of supervised classi cation [11]. Statistical analysis ... |

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Citation Context ...wn independently. We conclude in Section 5 with a number of directions for further work. 1.2 Related work Analysis of the structure of the Web graph has been used to enhance the quality of Web search =-=[5, 6, 8, 9, 21, 29]-=-. The topics of pages pointed to by aWeb page can be used to improve the accuracy of determining the (unknown) topic of this page in the setting of supervised classi cation [11]. Statistical analysis ... |

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Citation Context ...na in citation� some of these insights have been applied to the Web as well [25]. A view of the Web as a semi-structured database has been advanced by many authors. In particular, LORE [1] and WebSQL =-=[27]-=- use graph-theoretic and relational views of the Web respectively. These views support structured query interfaces to the Web (Lorel [1] and WebSQL [27]) that are evocative of and similar to SQL. An a... |

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Citation Context ...ted graph induced by the hyperlinks between Web pages� we refer to this as the Web graph. For our purposes, nodes represent static html pages and hyperlinks represent directed edges. Recent estimates =-=[4]-=- suggest that there are several hundred million nodes in the Web graph� this quantity isgrowing by a few percent a month. The average node has roughly seven hyperlinks (directed edges) to other pages,... |

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Citation Context ... numbers of \items" (pages) in the Web graph. This number is already two to three orders of magnitude more than the number of items in a typical market basket analysis. The work of Mendelzon and Wood =-=[28]-=- is an instance of structural methods in mining. They argue that the traditional SQL query interface to databases is inadequate in its power to specify several structural queries that are interesting ... |

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Citation Context ...dges) to other pages, making for a total of several billion hyperlinks in all.sThere are several reasons for studying the Web graph. The structure of this graph has already led to improved Web search =-=[6,8, 21, 29]-=-, more accurate topic-classi cation algorithms [11] and has inspired algorithms for enumerating emergent cyber-communities [23]. The hyperlinks themselves represent afecund source of sociological info... |

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Citation Context ...in spirit to our proposal, though di erent in details and application. The eld of bibliometrics [14, 16] has studied phenomena in citation� some of these insights have been applied to the Web as well =-=[25]-=-. A view of the Web as a semi-structured database has been advanced by many authors. In particular, LORE [1] and WebSQL [27] use graph-theoretic and relational views of the Web respectively. These vie... |

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Citation Context ...ements we have madeonthe entire Web graph, and on particular local subgraphs of interest. We show, for instance, that the in- and out-degrees of nodes follow Zip an (inverse polynomial) distributions =-=[12, 17, 26, 31]-=-. This and other measurements of the frequency of occurrence of certain structures suggest that traditional random graph models such asG n�p [7] are likely to do a poor job of modeling the Web graph. ... |

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Citation Context ...ements we have madeonthe entire Web graph, and on particular local subgraphs of interest. We show, for instance, that the in- and out-degrees of nodes follow Zip an (inverse polynomial) distributions =-=[12, 17, 26, 31]-=-. This and other measurements of the frequency of occurrence of certain structures suggest that traditional random graph models such asG n�p [7] are likely to do a poor job of modeling the Web graph. ... |

21 |
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Citation Context ... structure of this graph has already led to improved Web search [6,8, 21, 29], more accurate topic-classi cation algorithms [11] and has inspired algorithms for enumerating emergent cyber-communities =-=[23]-=-. The hyperlinks themselves represent afecund source of sociological information. Beyond the intrinsic interest of the structure of the Web graph, measurements of the graph and of the behavior of user... |

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Citation Context ...orithms for nding association rules and related statistical correlation measures in a given dataset. However, e cient methods such as a priori [2] or even more general methodologies such asquery ocks =-=[30]-=-, do not scale to the numbers of \items" (pages) in the Web graph. This number is already two to three orders of magnitude more than the number of items in a typical market basket analysis. The work o... |

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Citation Context ...ements we have madeonthe entire Web graph, and on particular local subgraphs of interest. We show, for instance, that the in- and out-degrees of nodes follow Zip an (inverse polynomial) distributions =-=[12, 17, 26, 31]-=-. This and other measurements of the frequency of occurrence of certain structures suggest that traditional random graph models such asG n�p [7] are likely to do a poor job of modeling the Web graph. ... |

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Citation Context ...Lotka in 1926 [26].sGilbert [17] presents a probabilistic model explaining Lotka's law, which issimilar in spirit to our proposal, though di erent in details and application. The eld of bibliometrics =-=[14, 16]-=- has studied phenomena in citation� some of these insights have been applied to the Web as well [25]. A view of the Web as a semi-structured database has been advanced by many authors. In particular, ... |

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Citation Context ...hyperlinks in all.sThere are several reasons for studying the Web graph. The structure of this graph has already led to improved Web search [6,8, 21, 29], more accurate topic-classi cation algorithms =-=[11]-=- and has inspired algorithms for enumerating emergent cyber-communities [23]. The hyperlinks themselves represent afecund source of sociological information. Beyond the intrinsic interest of the struc... |

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Citation Context ...ing to three pages would require examining approximately 10 40 possibilities on a graph with 10 8 nodes. A theoretical question (open as far as we know): does the work on xed-parameter intractability =-=[13]-=- imply that we cannot { in the worst case { improve on naive enumeration for bipartite cores? Such a result would argue that algorithms that are provably e cient ontheWeb graph must exploit some featu... |