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 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 .
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. Giant component appears also in percolation of lattice networks.
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 has degree , then the giant component exists if and only if
a. out-component is a set of vertices that can be reached by recursively following all out-edges forward;
b. in-component is a set of vertices that can be reached by recursively following all in-edges backward;
c. weak component is a set of vertices that can be reached by recursively following all edges regardless of their direction.
Let a randomly chosen vertex has in-edges and out edges. By definition, the average number of in- and out-edges coincides so that . The criteria for giant component existence in directed and undirected random graphs are given in the table below.
|undirected: giant component|||
|directed: giant in/out-component|||
|directed: giant weak component|||
- Graph theory
- Interdependent networks
- Percolation theory
- Phase transitions
- Complex Networks
- Network Science
- Scale free networks
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