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| In [[network theory]], a '''giant component''' is a [[Connected component (graph theory)|connected component]] of a given [[random graph]] that contains a constant fraction of the entire graph's [[vertex (graph theory)|vertices]].
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| Giant components are a prominent feature of the [[Erdős–Rényi model]] of random graphs, in which each possible edge connecting pairs of a given set of {{mvar|n}} vertices is present, independently of the other edges, with probability {{mvar|p}}. In this model, if <math>p \le \frac{1-\epsilon}{n}</math> for any constant <math>\epsilon>0</math>, then with high probability all connected components of the graph have size {{math|O(log ''n'')}}, and there is no giant component. However, for <math>p \ge \frac{1 + \epsilon}{n}</math> there is with high probability a single giant component, with all other components having size {{math|O(log ''n'')}}. For <math>p = \frac{1}{n}</math>, intermediate between these two possibilities, the number of vertices in the largest component of the graph is with high probability proportional to <math>n^{2/3}</math>.<ref name="b">{{citation|contribution=6. The Evolution of Random Graphs—The Giant Component|pages=130–159|title=Random Graphs|volume=73|series=Cambridge studies in advanced mathematics|first=Béla|last=Bollobás|edition=2nd|publisher=Cambridge University Press|year=2001|isbn=978-0-521-79722-1}}.</ref>
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| Alternatively, if one adds randomly selected edges one at a time, starting with an [[empty graph]], then it is not until approximately <math>n/2</math> edges have been added that the graph contains a large component, and soon after that the component becomes giant. More precisely, when <math>t</math> edges have been added, for values of <math>t</math> close to but larger than <math>n/2</math>, the size of the giant component is approximately <math>4t-2n</math>.<ref name="b"/> However, according to the [[coupon collector's problem]], <math>\Theta(n\log n)</math> edges are needed in order to have high probability that the whole random graph is connected.
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| A similar sharp threshold between parameters that lead to graphs with all components small and parameters that lead to a giant component also occurs in random graphs with non-uniform [[degree distribution]]s.<ref>{{citation|pages=113–142|contribution=6. The Rise of the Giant Component|title=Complex Graphs and Networks|volume=107|series=Regional Conference Series in Mathematics|first1=Fan R. K.|last1=Chung|author1-link=Fan Chung|first2=Linyuan Lu|last2=|publisher=American Mathematical Society|year=2006|isbn=978-0-8218-3657-6}}.</ref>
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| ==References==
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| {{reflist}}
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| {{Combin-stub}}
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| [[Category:Graph connectivity]]
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| [[Category:Random graphs]]
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Latest revision as of 11:38, 6 January 2015
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