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Giant component : ウィキペディア英語版
Giant component
In network theory, a giant component is a connected component of a given random graph that contains a constant fraction of the entire graph's vertices.
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 vertices is present, independently of the other edges, with probability . In this model, if p \le \frac for any constant \epsilon>0, then with high probability all connected components of the graph have size , and there is no giant component. However, for p \ge \frac there is with high probability a single giant component, with all other components having size . For p = \frac, intermediate between these two possibilities, the number of vertices in the largest component of the graph is with high probability proportional to n^.〔.〕
Alternatively, if one adds randomly selected edges one at a time, starting with an empty graph, then it is not until approximately n/2 edges have been added that the graph contains a large component, and soon after that the component becomes giant. More precisely, when t edges have been added, for values of t close to but larger than n/2, the size of the giant component is approximately 4t-2n.〔 However, according to the coupon collector's problem, \Theta(n\log n) edges are needed in order to have high probability that the whole random graph is connected.
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 distributions.〔.〕
==References==



抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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