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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 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 Window 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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 DiegoUSA
  2. 2.University of South CarolinaUSA

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