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 for any constant , then with high probability all connected components of the graph have size O(log n), and there is no giant component. However, for there is with high probability a single giant component, with all other components having size O(log n). For , intermediate between these two possibilities, the number of vertices in the largest component of the graph is with high probability proportional to .
Alternatively, if one adds randomly selected edges one at a time, starting with an empty graph, then it is not until approximately edges have been added that the graph contains a large component, and soon after that the component becomes giant. More precisely, when edges have been added, for values of close to but larger than , the size of the giant component is approximately . However, according to the coupon collector's problem, 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.
- 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.
- 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.
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