### Abstract

Preferential attachment networks with power law exponent τ> 3 are known to exhibit a phase transition. There is a value ρ
_{c}> 0 such that, for small edge densities ρ≤ ρ
_{c} every component of the graph comprises an asymptotically vanishing proportion of vertices, while for large edge densities ρ> ρ
_{c} there is a unique giant component comprising an asymptotically positive proportion of vertices. In this paper we study the decay in the size of the giant component as the critical edge density is approached from above. We show that the size decays very rapidly, like exp(-c/ρ-ρc) for an explicit constant c> 0 depending on the model implementation. This result is in contrast to the behaviour of the class of rank-one models of scale-free networks, including the configuration model, where the decay is polynomial. Our proofs rely on the local neighbourhood approximations of Dereich and Mörters (Ann Probab 41(1):329–384, 2013) and recent progress in the theory of branching random walks (Gantert et al. in Ann Inst Henri Poincaré Probab Stat 47(1):111–129, 2011).

Original language | English |
---|---|

Pages (from-to) | 663-703 |

Number of pages | 41 |

Journal | Journal of Statistical Physics |

Volume | 173 |

Issue number | 3-4 |

Early online date | 14 May 2018 |

DOIs | |

Publication status | Published - 1 Nov 2018 |

### Fingerprint

### Keywords

- Barabási-Albert model
- Killed branching random walk
- Percolation
- Preferential attachment
- Scale-free network
- Survival probability

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*173*(3-4), 663-703. https://doi.org/10.1007/s10955-018-2054-5

**Near critical preferential attachment networks have small giant components.** / Eckhoff, Maren; Morters, Peter; Ortgiese, Marcel.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 173, no. 3-4, pp. 663-703. https://doi.org/10.1007/s10955-018-2054-5

}

TY - JOUR

T1 - Near critical preferential attachment networks have small giant components

AU - Eckhoff, Maren

AU - Morters, Peter

AU - Ortgiese, Marcel

PY - 2018/11/1

Y1 - 2018/11/1

N2 - Preferential attachment networks with power law exponent τ> 3 are known to exhibit a phase transition. There is a value ρ c> 0 such that, for small edge densities ρ≤ ρ c every component of the graph comprises an asymptotically vanishing proportion of vertices, while for large edge densities ρ> ρ c there is a unique giant component comprising an asymptotically positive proportion of vertices. In this paper we study the decay in the size of the giant component as the critical edge density is approached from above. We show that the size decays very rapidly, like exp(-c/ρ-ρc) for an explicit constant c> 0 depending on the model implementation. This result is in contrast to the behaviour of the class of rank-one models of scale-free networks, including the configuration model, where the decay is polynomial. Our proofs rely on the local neighbourhood approximations of Dereich and Mörters (Ann Probab 41(1):329–384, 2013) and recent progress in the theory of branching random walks (Gantert et al. in Ann Inst Henri Poincaré Probab Stat 47(1):111–129, 2011).

AB - Preferential attachment networks with power law exponent τ> 3 are known to exhibit a phase transition. There is a value ρ c> 0 such that, for small edge densities ρ≤ ρ c every component of the graph comprises an asymptotically vanishing proportion of vertices, while for large edge densities ρ> ρ c there is a unique giant component comprising an asymptotically positive proportion of vertices. In this paper we study the decay in the size of the giant component as the critical edge density is approached from above. We show that the size decays very rapidly, like exp(-c/ρ-ρc) for an explicit constant c> 0 depending on the model implementation. This result is in contrast to the behaviour of the class of rank-one models of scale-free networks, including the configuration model, where the decay is polynomial. Our proofs rely on the local neighbourhood approximations of Dereich and Mörters (Ann Probab 41(1):329–384, 2013) and recent progress in the theory of branching random walks (Gantert et al. in Ann Inst Henri Poincaré Probab Stat 47(1):111–129, 2011).

KW - Barabási-Albert model

KW - Killed branching random walk

KW - Percolation

KW - Preferential attachment

KW - Scale-free network

KW - Survival probability

UR - http://www.scopus.com/inward/record.url?scp=85056568533&partnerID=8YFLogxK

U2 - 10.1007/s10955-018-2054-5

DO - 10.1007/s10955-018-2054-5

M3 - Article

VL - 173

SP - 663

EP - 703

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3-4

ER -