The Giant Component in a Random Subgraph of a Given Graph

  • Fan Chung
  • Paul Horn
  • Linyuan Lu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5427)

Abstract

We consider a random subgraph Gp 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 dv 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 Gp has a giant component. In the other direction, if \(p < (1-\epsilon)/\tilde{d}\) then almost surely the percolated subgraph Gp contains no giant component.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Fan Chung
    • 1
  • Paul Horn
    • 1
  • Linyuan Lu
    • 2
  1. 1.University of CaliforniaSan Diego 
  2. 2.University of South Carolina 

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