# Giant component

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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, $P_{inf}$ 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 depends 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 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. This example can be generalized to a system of n ER networks connected via dependency links in a tree like structure. Interestingly for any tree formed of n ER interdependent 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, $P_{inf}=p(1-\exp(-\langle k\rangle P_{inf}))^{n}$ .

## Reinforced nodes

The percolation giant component in the presence of reinforced (decentralization of the network) has been studied by Yuan et al. 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 $k$ , 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.