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 for any constant , then with high probability all connected components of the graph have size O(log n), and there is no giant component. However, for there is with high probability a single giant component, with all other components having size O(log n). For , intermediate between these two possibilities, the number of vertices in the largest component of the graph, is with high probability proportional to .[1]
Giant component is also important in percolation theory.[1][2][3][4] When a fraction of nodes, , is removed randomly from an ER network of degree , there exists a critical threshold, . Above there exists a giant component (largest cluster) of size, . fulfills, . For the solution of this equation is , i.e., there is no giant component.
At , the distribution of cluster sizes behaves as a power law, ~ 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 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 , the size of the giant component is approximately .[1] However, according to the coupon collector's problem, 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, which is a natural generalization of the formula for a single network, .
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 , then the giant component exists[8] if and only if
- out-component is a set of vertices that can be reached by recursively following all out-edges forward;
- in-component is a set of vertices that can be reached by recursively following all in-edges backward;
- 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 in-edges and out edges. By definition, the average number of in- and out-edges coincides so that . If is the generating function of the degree distribution for an undirected network, then can be defined as . For directed networks, generating function assigned to the joint probability distribution can be written with two valuables and as: , then one can define and . The criteria for giant component existence in directed and undirected random graphs are given in the table below:
Type | Criteria |
---|---|
undirected: giant component | ,[8] or [9] |
directed: giant in/out-component | ,[9] or [9] |
directed: giant weak component | [10] |
For other properties of the giant component and its relation to percolation theory and critical phenomena, see references.[3][4][2]
See also[]
- Erdős–Rényi model
- Fractals
- Graph theory
- Interdependent networks
- Percolation theory
- Percolation
- Complex Networks
- Network Science
- Scale free networks
References[]
- ^ Jump up to: 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.
- ^ Jump up to: a b c Armin, Bunde (1996). Fractals and Disordered Systems. Havlin, Shlomo. (Second Revision, Enlarged ed.). Berlin, Heidelberg: Springer Berlin Heidelberg. ISBN 9783642848681. OCLC 851388749.
- ^ Jump up to: a b Cohen, Reuven; Havlin, Shlomo (2010). Complex Networks: Structure, Robustness and Function. Cambridge: Cambridge University Press. doi:10.1017/cbo9780511780356. ISBN 9780521841566.
- ^ Jump up to: a b Newman, M. E. J. (2010). Networks : an introduction. New York: Oxford University Press. OCLC 456837194.
- ^ Jump up to: a b Buldyrev, Sergey V.; Parshani, Roni; Paul, Gerald; Stanley, H. Eugene; Havlin, Shlomo (2010). "Catastrophic cascade of failures in interdependent networks". Nature. 464 (7291): 1025–1028. arXiv:0907.1182. Bibcode:2010Natur.464.1025B. doi:10.1038/nature08932. ISSN 0028-0836. PMID 20393559. S2CID 1836955.
- ^ Gao, Jianxi; Buldyrev, Sergey V.; Stanley, H. Eugene; Havlin, Shlomo (2011-12-22). "Networks formed from interdependent networks". Nature Physics. 8 (1): 40–48. doi:10.1038/nphys2180. ISSN 1745-2473.
- ^ X. Yuan, Y. Hu, H.E. Stanley, S. Havlin (2017). "Eradicating catastrophic collapse in interdependent networks via reinforced nodes". PNAS. 114 (13): 3311–3315. doi:10.1073/pnas.1621369114. PMC 5380073. PMID 28289204.CS1 maint: multiple names: authors list (link)
- ^ Jump up to: a b Molloy, Michael; Reed, Bruce (1995). "A critical point for random graphs with a given degree sequence". Random Structures & Algorithms. 6 (2–3): 161–180. doi:10.1002/rsa.3240060204. ISSN 1042-9832.
- ^ Jump up to: a b c d Newman, M. E. J.; Strogatz, S. H.; Watts, D. J. (2001-07-24). "Random graphs with arbitrary degree distributions and their applications". Physical Review E. 64 (2): 026118. arXiv:cond-mat/0007235. Bibcode:2001PhRvE..64b6118N. doi:10.1103/physreve.64.026118. ISSN 1063-651X. PMID 11497662.
- ^ Kryven, Ivan (2016-07-27). "Emergence of the giant weak component in directed random graphs with arbitrary degree distributions". Physical Review E. 94 (1): 012315. arXiv:1607.03793. Bibcode:2016PhRvE..94a2315K. doi:10.1103/physreve.94.012315. ISSN 2470-0045. PMID 27575156. S2CID 206251373.
- Graph connectivity
- Random graphs