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 .
Giant component is also important in percolation theory. 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.
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 edges have been added, for values of close to but larger than , the size of the giant component is approximately . 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. 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, which is a natural generalization of the formula for a single network, .
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 , then the giant component exists 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:
|undirected: giant component||, or |
|directed: giant in/out-component||, or |
|directed: giant weak component|||
- Erdős–Rényi model
- Graph theory
- Interdependent networks
- Percolation theory
- Complex Networks
- Network Science
- Scale free networks
- 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.
- Armin, Bunde (1996). Fractals and Disordered Systems. Havlin, Shlomo. (Second Revision, Enlarged ed.). Berlin, Heidelberg: Springer Berlin Heidelberg. ISBN 9783642848681. OCLC 851388749.
- Cohen, Reuven; Havlin, Shlomo (2010). Complex Networks: Structure, Robustness and Function. Cambridge: Cambridge University Press. doi:10.1017/cbo9780511780356. ISBN 9780521841566.
- Newman, M. E. J. (2010). Networks : an introduction. New York: Oxford University Press. OCLC 456837194.
- 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.
- 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: 3311.CS1 maint: multiple names: authors list (link)
- 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.
- 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.