Giant component

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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 n vertices is present, independently of the other edges, with probability p. In this model, if p \le \frac{1-\epsilon}{n} for any constant \epsilon>0, then with high probability all connected components of the graph have size O(log n), and there is no giant component. However, for p \ge \frac{1 + \epsilon}{n} there is with high probability a single giant component, with all other components having size O(log n). For p = \frac{1}{n}, intermediate between these two possibilities, the number of vertices in the largest component of the graph is with high probability proportional to n^{2/3}.[1]

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.[1] 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.[2]

References[edit]

  1. ^ a b Bollobás, Béla (2001), "6. The Evolution of Random Graphs—The Giant Component", Random Graphs, Cambridge studies in advanced mathematics 73 (2nd ed.), Cambridge University Press, pp. 130–159, ISBN 978-0-521-79722-1 .
  2. ^ Chung, Fan R. K. (2006), "6. The Rise of the Giant Component", Complex Graphs and Networks, Regional Conference Series in Mathematics 107, American Mathematical Society, pp. 113–142, ISBN 978-0-8218-3657-6  |first2= missing |last2= in Authors list (help).