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 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 $n^{2/3}$.

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.