### 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 d_{v} becomes active with probability α(d_{v}) 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(d_{v}) 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 language | English (US) |
---|---|

Pages (from-to) | 1-20 |

Number of pages | 20 |

Journal | Electronic Journal of Combinatorics |

Volume | 17 |

Issue number | 1 |

State | Published - 2010 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Geometry and Topology
- Theoretical Computer Science
- Computational Theory and Mathematics

### Cite this

**Bootstrap percolation and diffusion in random graphs with given vertex degrees.** / Amini, Leo Hamed.

Research output: Contribution to journal › Article

*Electronic Journal of Combinatorics*, vol. 17, no. 1, pp. 1-20.

}

TY - JOUR

T1 - Bootstrap percolation and diffusion in random graphs with given vertex degrees

AU - Amini, Leo Hamed

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

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UR - http://www.scopus.com/inward/citedby.url?scp=77955591135&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:77955591135

VL - 17

SP - 1

EP - 20

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 1

ER -