# 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 $p\leq {\frac {1-\epsilon }{n}}$  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 $p\geq {\frac {1+\epsilon }{n}}$  there is with high probability a single giant component, with all other components having size O(log n). For $p=p_{c}={\frac {1}{n}}$ , intermediate between these two possibilities, the number of vertices in the largest component of the graph is with high probability proportional to $n^{2/3}$ .

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

At $p_{c}$ , the distribution of cluster sizes behaves as a power law, $n(s)$ ~$s^{-5/2}$  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 $n/2$  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.

## Giant component in interdependent networks

Consider for simplicity two ER networks with same number of nodes and same degree. Each node in one network depend on a node (for functioning) in the other network and vice versa through bi-directional links. This system is called interdependent networks. In order for system to survive removals, both networks should have giant components where each node in one depend on a node in the other. This is called the mutual giant component. This example can be generalized to n ER networks connected in a tree like structure. Interestingly for any tree of n ER networks the mutual giant component (MGC) is given by, $P_{inf}=p(1-\exp(-\langle k\rangle P_{inf})^{n}$  which is a natural generalization of the formula for a single network.

## 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 $k$ , then the giant component exists if and only if

$\mathbb {E} [k^{2}]-2\mathbb {E} [k]>0.$

$\mathbb {E} [k]$  which is also written in the form of ${\langle k\rangle }$  is the mean degree of the network. 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:
1. out-component is a set of vertices that can be reached by recursively following all out-edges forward;
2. in-component is a set of vertices that can be reached by recursively following all in-edges backward;
3. weak component is a set of vertices that can be reached by recursively following all edges regardless of their direction.

## Criteria for giant component existence in directed and undirected configuration graphs

Let a randomly chosen vertex has $k_{\text{in}}$  in-edges and $k_{\text{out}}$  out edges. By definition, the average number of in- and out-edges coincides so that $c=\mathbb {E} [k_{\text{in}}]=\mathbb {E} [k_{\text{out}}]$ . If $G_{0}(x)=\textstyle \sum _{k}\displaystyle P(k)x^{k}$  is the generating function of the degree distribution $P(k)$  for an undirected network, then $G_{1}(x)$  can be defined as $G_{1}(x)=\textstyle \sum _{k}\displaystyle {\frac {k}{\langle k\rangle }}P(k)x^{k-1}$ . For directed networks, generating function assigned to the joint probability distribution $P(k_{in},k_{out})$  can be written with two valuables $x$  and $y$  as: ${\mathcal {G}}(x,y)=\sum _{k_{in},k_{out}}\displaystyle P({k_{in},k_{out}})x^{k_{in}}y^{k_{out}}$ , then one can define $g(x)={\frac {1}{c}}{\partial {\mathcal {G}} \over \partial x}\vert _{y=1}$  and $f(y)={\frac {1}{c}}{\partial {\mathcal {G}} \over \partial y}\vert _{x=1}$ . The criteria for giant component existence in directed and undirected random graphs are given in the table below:

Type Criteria
undirected: giant component $\mathbb {E} [k^{2}]-2\mathbb {E} [k]>0$ , or $G'_{1}(1)=1$ 
directed: giant in/out-component $\mathbb {E} [k_{\text{in}}k_{out}]-\mathbb {E} [k_{\text{in}}]>0$ , or $g'_{1}(1)=f'_{1}(1)=1$ 
directed: giant weak component $2\mathbb {E} [k_{\text{in}}]\mathbb {E} [k_{\text{in}}k_{\text{out}}]-\mathbb {E} [k_{\text{in}}]\mathbb {E} [k_{\text{out}}^{2}]-\mathbb {E} [k_{\text{in}}]\mathbb {E} [k_{\text{in}}^{2}]+\mathbb {E} [k_{\text{in}}^{2}]\mathbb {E} [k_{\text{out}}^{2}]-\mathbb {E} [k_{\text{in}}k_{\text{out}}]^{2}>0$ 

For other properties of the giant component and its relation to percolation theory and critical phenomena, see references.