Bootstrap percolation and diffusion in random graphs with given vertex degrees

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Abstract

We consider diffusion in random graphs with given vertex degrees. Our diffusion model can be viewed as a variant of a cellular automaton growth process: assume that each node can be in one of the two possible states, inactive or active. The parameters of the model are two given functions θ: N → N and α:N → [0,1]. At the beginning of the process, each node v of degree dv becomes active with probability α(dv) independently of the other vertices. Presence of the active vertices triggers a percolation process: if a node v is active, it remains active forever. And if it is inactive, it will become active when at least θ(dv) of its neighbors are active. In the case where α(d) = α and θ(d) = θ, for each d ∈ N, our diffusion model is equivalent to what is called bootstrap percolation. The main result of this paper is a theorem which enables us to find the final proportion of the active vertices in the asymptotic case, i.e., when n → ∞. This is done via analysis of the process on the multigraph counterpart of the graph model.

Original languageEnglish (US)
Pages (from-to)1-20
Number of pages20
JournalElectronic Journal of Combinatorics
Volume17
Issue number1
StatePublished - Aug 20 2010
Externally publishedYes

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ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics
  • Applied Mathematics

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