# 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, ${\displaystyle P_{\inf }}$  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, ${\displaystyle n(s)}$ ~${\displaystyle s^{-5/2}}$  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 t edges have been added, for values of 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.

## Giant component in interdependent networks

Consider for simplicity two ER networks with same number of nodes and same degree. Each node in one network depends on a node (for functioning) in the other network and vice versa through bi-directional links. This system is called interdependent networks.[5] In order for the system to function, both networks should have giant components where each node in one network depends on a node in the other. This is called the mutual giant component.[5] This example can be generalized to a system of n ER networks connected via dependency links in a tree like structure.[6] Interestingly for any tree formed of n ER interdependent networks, the mutual giant component (MGC) is given by, ${\displaystyle P_{\inf }=p(1-\exp(-\langle k\rangle P_{\inf }))^{n}}$  which is a natural generalization of the formula for a single network, ${\displaystyle P_{\inf }=p(1-\exp(-\langle k\rangle P_{\inf }))^{n}}$ .

## Reinforced nodes

The percolation giant component in the presence of reinforced (decentralization of the network) has been studied by Yuan et al.[7] Reinforced nodes have extra sources that can support the finite components in which they belong , i.e., equivalent to having alternative links to the giant components.

## 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 ${\displaystyle k}$ , then the giant component exists[8] if and only if

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

${\displaystyle \mathbb {E} [k]}$  which is also written in the form of ${\displaystyle {\langle k\rangle }}$  is the mean degree of the network. Similar expressions are also valid for directed graphs, in which case the degree distribution is two-dimensional.[9] 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 ${\displaystyle k_{\text{in}}}$  in-edges and ${\displaystyle k_{\text{out}}}$  out edges. By definition, the average number of in- and out-edges coincides so that ${\displaystyle c=\mathbb {E} [k_{\text{in}}]=\mathbb {E} [k_{\text{out}}]}$ . If ${\displaystyle G_{0}(x)=\textstyle \sum _{k}\displaystyle P(k)x^{k}}$  is the generating function of the degree distribution ${\displaystyle P(k)}$  for an undirected network, then ${\displaystyle G_{1}(x)}$  can be defined as ${\displaystyle G_{1}(x)=\textstyle \sum _{k}\displaystyle {\frac {k}{\langle k\rangle }}P(k)x^{k-1}}$ . For directed networks, generating function assigned to the joint probability distribution ${\displaystyle P(k_{in},k_{out})}$  can be written with two valuables ${\displaystyle x}$  and ${\displaystyle y}$  as: ${\displaystyle {\mathcal {G}}(x,y)=\sum _{k_{in},k_{out}}\displaystyle P({k_{in},k_{out}})x^{k_{in}}y^{k_{out}}}$ , then one can define ${\displaystyle g(x)={\frac {1}{c}}{\partial {\mathcal {G}} \over \partial x}\vert _{y=1}}$  and ${\displaystyle f(y)={\frac {1}{c}}{\partial {\mathcal {G}} \over \partial y}\vert _{x=1}}$ . The criteria for giant component existence in directed and undirected random graphs are given in the table below:

Type Criteria
undirected: giant component ${\displaystyle \mathbb {E} [k^{2}]-2\mathbb {E} [k]>0}$ ,[8] or ${\displaystyle G'_{1}(1)=1}$ [9]
directed: giant in/out-component ${\displaystyle \mathbb {E} [k_{\text{in}}k_{out}]-\mathbb {E} [k_{\text{in}}]>0}$ ,[9] or ${\displaystyle g'_{1}(1)=f'_{1}(1)=1}$ [9]
directed: giant weak component ${\displaystyle 2\mathbb {E} [k_{\text{in}}]\mathbb {E} [k_{\text{in}}k_{\text{out}}]-\mathbb {E} [k_{\text{in}}]\mathbb {E} [k_{\text{out}}^{2}]-\mathbb {E} [k_{\text{in}}]\mathbb {E} [k_{\text{in}}^{2}]+\mathbb {E} [k_{\text{in}}^{2}]\mathbb {E} [k_{\text{out}}^{2}]-\mathbb {E} [k_{\text{in}}k_{\text{out}}]^{2}>0}$ [10]

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