# Giant component

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

Giant component is also important in percolation theory.[1][2][3][4] When a fraction of nodes, ${\displaystyle q=1-p}$, is removed randomly from an ER network of degree ${\displaystyle \langle k\rangle }$, there exists a critical threshold, ${\displaystyle p_{c}={\frac {1}{\langle k\rangle }}}$. Above ${\displaystyle p_{c}}$ there exists a giant component (largest cluster) of size, ${\displaystyle P_{inf}}$. ${\displaystyle P_{inf}}$ fulfills, ${\displaystyle P_{inf}=p(1-\exp(-\langle k\rangle P_{inf})}$. For ${\displaystyle p the solution of this equation is ${\displaystyle P_{inf}=0}$, i.e., there is no giant component.

At ${\displaystyle p_{c}}$, the distribution of cluster sizes behaves as a power law, ${\displaystyle n(s)~s^{-5/2}}$ which is a feature of phase transition. Giant component appears also in percolation of lattice networks.[2]

Alternatively, if one adds randomly selected edges one at a time, starting with an empty graph, then it is not until approximately ${\displaystyle n/2}$ edges have been added that the graph contains a large component, and soon after that the component becomes giant. More precisely, when ${\displaystyle t}$ edges have been added, for values of ${\displaystyle t}$ close to but larger than ${\displaystyle n/2}$, the size of the giant component is approximately ${\displaystyle 4t-2n}$.[1] However, according to the coupon collector's problem, ${\displaystyle \Theta (n\log n)}$ edges are needed in order to have high probability that the whole random graph is connected.

## Graphs with arbitrary degree distribution

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 ${\displaystyle k}$, then the giant component exists[5] if and only if

${\displaystyle \mathbb {E} [k^{2}]-2\mathbb {E} [k]>0.}$
Similar expressions are also valid for directed graphs, in which case the degree distribution is two-dimensional. There are three types of connected components in directed graphs. For a randomly chosen vertex:

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 ${\displaystyle k_{\text{in}}}$ in-edges and ${\displaystyle k_{\text{in}}}$ out edges. By definition, the average number of in- and out-edges coincides so that ${\displaystyle \mathbb {E} [k_{\text{in}}]=\mathbb {E} [k_{\text{out}}]}$. The criteria for giant component existence in directed and undirected random graphs are given in the table below.

Type Criteria
undirected: giant component ${\displaystyle \mathbb {E} [k^{2}]-2\mathbb {E} [k]>0}$[5]
directed: giant in/out-component ${\displaystyle \mathbb {E} [k_{\text{in}}k_{out}]-\mathbb {E} [k_{\text{in}}]>0}$[6]
directed: giant weak component ${\displaystyle 2\mathbb {E} [k_{\text{in}}]\mathbb {E} [k_{\text{in}}k_{\text{out}}]-\mathbb {E} [k_{\text{in}}]\mathbb {E} [k_{\text{out}}^{2}]-\mathbb {E} [k_{\text{in}}]\mathbb {E} [k_{\text{in}}^{2}]+[k_{\text{in}}^{2}]k_{\text{out}}^{2}]-\mathbb {E} [k_{\text{in}}k_{\text{out}}]^{2}>0}$[7]
Criteria for giant component existence in directed and undirected configuration graphs, ${\displaystyle \mathbb {E} [k_{\text{in}}]=\mathbb {E} [k_{\text{out}}]}$

For other properties of the giant component and its relation to percolation theory and critical phenomena, see references.[3][4][2]