### Abstract

We study random subgraphs of an arbitrary finite connected transitive graph G obtained by independently deleting edges with probability 1 - p. Let V be the number of vertices in G, and let Ω be their degree. We define the critical threshold p _{c} = p _{c}(G, λ) to be the value of p for which the expected cluster size of a fixed vertex attains the value λV ^{1/3}, where λ is fixed and positive. We show that, for any such model, there is a phase transition at p _{c} analogous to the phase transition for the random graph, provided that a quantity called the triangle diagram is sufficiently small at the threshold p _{c}. In particular, we show that the largest cluster inside a scaling window of size |p-p _{c}| = Θ(Ω ^{-1} V ^{-1/3}) is of size Θ(V ^{2/3}), while, below this scaling window, it is much smaller, of order O(ε ^{2} log(Vε ^{-3})), with ε = Ω(p _{c} - p). We also obtain an upper bound O(Ω(p -p _{c})V) for the expected size of the largest cluster above the window. In addition, we define and analyze the percolation probability above the window and show that it is of order Θ(Ω(p - p _{c})). Among the models for which the triangle diagram is small enough to allow us to draw these conclusions are the random graph, the n-cube and certain Hamming cubes, as well as the spread-out n-dimensional torus for n > 6.

Original language | English (US) |
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Pages (from-to) | 137-184 |

Number of pages | 48 |

Journal | Random Structures and Algorithms |

Volume | 27 |

Issue number | 2 |

DOIs | |

State | Published - Sep 2005 |

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### ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design
- Software
- Mathematics(all)
- Applied Mathematics

### Cite this

*Random Structures and Algorithms*,

*27*(2), 137-184. https://doi.org/10.1002/rsa.20051

**Random subgraphs of finite graphs : I. The scaling window under the triangle condition.** / Borgs, Christian; Chayes, Jennifer T.; Van Der Hofstad, Remco; Slade, Gordon; Spencer, Joel.

Research output: Contribution to journal › Article

*Random Structures and Algorithms*, vol. 27, no. 2, pp. 137-184. https://doi.org/10.1002/rsa.20051

}

TY - JOUR

T1 - Random subgraphs of finite graphs

T2 - I. The scaling window under the triangle condition

AU - Borgs, Christian

AU - Chayes, Jennifer T.

AU - Van Der Hofstad, Remco

AU - Slade, Gordon

AU - Spencer, Joel

PY - 2005/9

Y1 - 2005/9

N2 - We study random subgraphs of an arbitrary finite connected transitive graph G obtained by independently deleting edges with probability 1 - p. Let V be the number of vertices in G, and let Ω be their degree. We define the critical threshold p c = p c(G, λ) to be the value of p for which the expected cluster size of a fixed vertex attains the value λV 1/3, where λ is fixed and positive. We show that, for any such model, there is a phase transition at p c analogous to the phase transition for the random graph, provided that a quantity called the triangle diagram is sufficiently small at the threshold p c. In particular, we show that the largest cluster inside a scaling window of size |p-p c| = Θ(Ω -1 V -1/3) is of size Θ(V 2/3), while, below this scaling window, it is much smaller, of order O(ε 2 log(Vε -3)), with ε = Ω(p c - p). We also obtain an upper bound O(Ω(p -p c)V) for the expected size of the largest cluster above the window. In addition, we define and analyze the percolation probability above the window and show that it is of order Θ(Ω(p - p c)). Among the models for which the triangle diagram is small enough to allow us to draw these conclusions are the random graph, the n-cube and certain Hamming cubes, as well as the spread-out n-dimensional torus for n > 6.

AB - We study random subgraphs of an arbitrary finite connected transitive graph G obtained by independently deleting edges with probability 1 - p. Let V be the number of vertices in G, and let Ω be their degree. We define the critical threshold p c = p c(G, λ) to be the value of p for which the expected cluster size of a fixed vertex attains the value λV 1/3, where λ is fixed and positive. We show that, for any such model, there is a phase transition at p c analogous to the phase transition for the random graph, provided that a quantity called the triangle diagram is sufficiently small at the threshold p c. In particular, we show that the largest cluster inside a scaling window of size |p-p c| = Θ(Ω -1 V -1/3) is of size Θ(V 2/3), while, below this scaling window, it is much smaller, of order O(ε 2 log(Vε -3)), with ε = Ω(p c - p). We also obtain an upper bound O(Ω(p -p c)V) for the expected size of the largest cluster above the window. In addition, we define and analyze the percolation probability above the window and show that it is of order Θ(Ω(p - p c)). Among the models for which the triangle diagram is small enough to allow us to draw these conclusions are the random graph, the n-cube and certain Hamming cubes, as well as the spread-out n-dimensional torus for n > 6.

UR - http://www.scopus.com/inward/record.url?scp=26944446964&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=26944446964&partnerID=8YFLogxK

U2 - 10.1002/rsa.20051

DO - 10.1002/rsa.20051

M3 - Article

AN - SCOPUS:26944446964

VL - 27

SP - 137

EP - 184

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 2

ER -