The Giant Component in a Random Subgraph of a Given Graph
Abstract
We consider a random subgraph G p of a host graph G formed by retaining each edge of G with probability p. We address the question of determining the critical value p (as a function of G) for which a giant component emerges. Suppose G satisfies some (mild) conditions depending on its spectral gap and higher moments of its degree sequence. We define the second order average degree \(\tilde{d}\) to be \(\tilde{d}=\sum_v d_v^2/(\sum_v d_v)\) where d v denotes the degree of v. We prove that for any ε> 0, if \(p > (1+ \epsilon)/{\tilde d}\) then asymptotically almost surely the percolated subgraph G p has a giant component. In the other direction, if \(p < (1-\epsilon)/\tilde{d}\) then almost surely the percolated subgraph G p contains no giant component.
Keywords
Adjacency Matrix Random Graph Degree Sequence Giant Component Critical WindowPreview
Unable to display preview. Download preview PDF.
References
- 1.Alon, N., Benjamini, I., Stacey, A.: Percolation on finite graphs and isoperimetric inequalities. Annals of Probability 32(3), 1727–1745 (2004)MathSciNetMATHGoogle Scholar
- 2.Ajtai, M., Komlós, J., Szemerédi, E.: Largest random component of a k-cube. Combinatorica 2, 1–7 (1982)MathSciNetCrossRefMATHGoogle Scholar
- 3.Bollobas, B., Kohayakawa, Y., Łuczak, T.: The evolution of random subgraphs of the cube. Random Structures and Algorithms 3(1), 55–90 (1992)MathSciNetCrossRefMATHGoogle Scholar
- 4.Bollobas, B., Borgs, C., Chayes, J., Riordan, O.: Percolation on dense graph sequences (preprint)Google Scholar
- 5.Borgs, C., Chayes, J., van der Hofstad, R., Slade, G., Spencer, J.: Random subgraphs of finite graphs. I. The scaling window under the triangle condition. Random Structures and Algorithms 27(2), 137–184 (2005)MathSciNetCrossRefMATHGoogle Scholar
- 6.Borgs, C., Chayes, J., van der Hofstad, R., Slade, G., Spencer, J.: Random subgraphs of finite graphs. III. The scaling window under the triangle condition. Combinatorica 26(4), 395–410 (2006)MathSciNetCrossRefMATHGoogle Scholar
- 7.Chung, F.: Spectral Graph Theory. AMS Publications (1997)Google Scholar
- 8.Chung, F., Lu, L., Vu, V.: The spectra of random graphs with given expected degrees. Internet Mathematics 1, 257–275 (2004)MathSciNetCrossRefMATHGoogle Scholar
- 9.Chung, F., Lu, L.: Connected components in random graphs with given expected degree sequences. Annals of Combinatorics 6, 125–145 (2002)MathSciNetCrossRefMATHGoogle Scholar
- 10.Chung, F., Lu, L.: Complex Graphs and Networks. AMS Publications (2006)Google Scholar
- 11.Erdős, P., Rényi, A.: On Random Graphs I. Publ. Math Debrecen 6, 290–297 (1959)MathSciNetMATHGoogle Scholar
- 12.Frieze, A., Krivelevich, M., Martin, R.: The emergence of a giant component of pseudo-random graphs. Random Structures and Algorithms 24, 42–50 (2004)MathSciNetCrossRefMATHGoogle Scholar
- 13.Grimmett, G.: Percolation. Springer, New York (1989)MATHGoogle Scholar
- 14.Kesten, H.: Percolation theory for mathematicians. In: Progress in Probability and Statistics, vol. 2, Birkhäuser, Boston (1982)Google Scholar
- 15.Malon, C., Pak, I.: Percolation on finite cayley graphs. In: Rolim, J.D.P., Vadhan, S.P. (eds.) RANDOM 2002. LNCS, vol. 2483, pp. 91–104. Springer, Heidelberg (2002)CrossRefGoogle Scholar
- 16.Nachmias, A.: Mean-field conditions for percolation in finite graphs (preprint, 2007)Google Scholar
- 17.Nachmias, A., Peres, Y.: Critical percolation on random regular graphs (preprint, 2007)Google Scholar