# 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 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[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]

## References

1. ^ a b c Bollobás, Béla (2001), "6. The Evolution of Random Graphs—The Giant Component", Random Graphs, Cambridge studies in advanced mathematics, 73 (2nd ed.), Cambridge University Press, pp. 130–159, ISBN 978-0-521-79722-1.
2. ^ a b c Armin., Bunde (1996). Fractals and Disordered Systems. Havlin, Shlomo. (Second rev. and enlarged ed.). Berlin, Heidelberg: Springer Berlin Heidelberg. ISBN 9783642848681. OCLC 851388749.
3. ^ a b Cohen, Reuven; Havlin, Shlomo (2010). Complex Networks: Structure, Robustness and Function. Cambridge: Cambridge University Press. doi:10.1017/cbo9780511780356. ISBN 9780521841566.
4. ^ a b Newman, M. E. J. (2010). Networks : an introduction. New York: Oxford University Press. OCLC 456837194.
5. ^ a b M. Molloy and B. Reed (1995). "A critical point for random graphs with a given degree sequence". "Random Struct. Algorithms" 6, 161
6. ^ M. E. J. Newman, S. H. Strogatz, and D. J. Watts (2001). "Random graphs with arbitrary degree distributions and their applications". Phys. Rev. E 64, 026118
7. ^ I. Kryven (2016). "Emergence of the giant weak component in directed random graphs with arbitrary degree distributions" Phys. Rev. E 94, 012315