In network theory, a giant component is a connected component of a given random graph that contains a finite fraction of the entire graph's vertices. Giant component in Erdős–Rényi model Giant components are a prominent feature of the Erdős–Rényi model (ER) 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 for any constant , then with high probability all connected components of the graph have size , and there is no giant component. However, for there is with high probability a single giant component, with all other components having size . For , intermediate between these two possibilities, the number of vertices in the largest component of the graph, is with high probability proportional to . Giant component is also important in percolation theory. When a fraction of nodes, , is removed randomly from an ER network of degree , there exists a critical threshold, . Above there exists a giant component (largest cluster) of size, . fulfills, . For the solution of this equation is , i.e., there is no giant component. At , the distribution of cluster sizes behaves as a power law, ~ which is a feature of phase transition. 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. Graphs with arbitrary degree distributions 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. The degree distribution does not define a graph uniquely. However under assumption that in all respects other than their degree distribution, the graphs are treated as entirely random, many results on finite/infinite-component sizes are known. In this model, the existence of the giant component depends only on the first two (mixed) moments of the degree distribution. Let a randomly chosen vertex