### Abstract

Critical phenomena in bootstrap percolation processes were originally observed by Aizenman and Lebowitz in the late 1980s as finite-volume phase transitions in Zd that are caused by the accumulation of small local islands of infected vertices. They were also observed in the case of dense (homogeneous) random graphs by Janson et al. [Ann. Appl. Probab. 22 (2012) 1989–2047]. In this paper, we consider the class of inhomogeneous random graphs known as the Chung-Lu model: each vertex is equipped with a positive weight and each pair of vertices appears as an edge with probability proportional to the product of the weights. In particular, we focus on the sparse regime, where the number of edges is proportional to the number of vertices.

The main results of this paper determine those weight sequences for which a critical phenomenon occurs: there is a critical density of vertices that are infected at the beginning of the process, above which a small (sublinear) set of infected vertices creates an avalanche of infections that in turn leads to an outbreak. We show that this occurs essentially only when the tail of the weight distribution dominates a power law with exponent 3 and we determine the critical density in this case.

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

Pages (from-to) | 990-1051 |

Journal | The annals of applied probability |

Volume | 28 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2018 |

### Fingerprint

### Fields of Expertise

- Information, Communication & Computing

### Cite this

*The annals of applied probability*,

*28*(2), 990-1051. https://doi.org/10.1214/17-AAP1324, https://doi.org/10.1214/17-AAP1324

**A phase transition regarding the evolution of bootstrap processes in inhomogeneous random graphs.** / Fountoulakis, Nikolaos; Kang, Mihyun; Koch, Christoph Jörg; Makai, Tamas.

Research output: Contribution to journal › Article › Research › peer-review

*The annals of applied probability*, vol. 28, no. 2, pp. 990-1051. https://doi.org/10.1214/17-AAP1324, https://doi.org/10.1214/17-AAP1324

}

TY - JOUR

T1 - A phase transition regarding the evolution of bootstrap processes in inhomogeneous random graphs

AU - Fountoulakis, Nikolaos

AU - Kang, Mihyun

AU - Koch, Christoph Jörg

AU - Makai, Tamas

PY - 2018

Y1 - 2018

N2 - A bootstrap percolation process on a graph with infection threshold r≥1 is a dissemination process that evolves in time steps. The process begins with a subset of infected vertices and in each subsequent step every uninfected vertex that has at least r infected neighbours becomes infected and remains so forever.Critical phenomena in bootstrap percolation processes were originally observed by Aizenman and Lebowitz in the late 1980s as finite-volume phase transitions in Zd that are caused by the accumulation of small local islands of infected vertices. They were also observed in the case of dense (homogeneous) random graphs by Janson et al. [Ann. Appl. Probab. 22 (2012) 1989–2047]. In this paper, we consider the class of inhomogeneous random graphs known as the Chung-Lu model: each vertex is equipped with a positive weight and each pair of vertices appears as an edge with probability proportional to the product of the weights. In particular, we focus on the sparse regime, where the number of edges is proportional to the number of vertices.The main results of this paper determine those weight sequences for which a critical phenomenon occurs: there is a critical density of vertices that are infected at the beginning of the process, above which a small (sublinear) set of infected vertices creates an avalanche of infections that in turn leads to an outbreak. We show that this occurs essentially only when the tail of the weight distribution dominates a power law with exponent 3 and we determine the critical density in this case.

AB - A bootstrap percolation process on a graph with infection threshold r≥1 is a dissemination process that evolves in time steps. The process begins with a subset of infected vertices and in each subsequent step every uninfected vertex that has at least r infected neighbours becomes infected and remains so forever.Critical phenomena in bootstrap percolation processes were originally observed by Aizenman and Lebowitz in the late 1980s as finite-volume phase transitions in Zd that are caused by the accumulation of small local islands of infected vertices. They were also observed in the case of dense (homogeneous) random graphs by Janson et al. [Ann. Appl. Probab. 22 (2012) 1989–2047]. In this paper, we consider the class of inhomogeneous random graphs known as the Chung-Lu model: each vertex is equipped with a positive weight and each pair of vertices appears as an edge with probability proportional to the product of the weights. In particular, we focus on the sparse regime, where the number of edges is proportional to the number of vertices.The main results of this paper determine those weight sequences for which a critical phenomenon occurs: there is a critical density of vertices that are infected at the beginning of the process, above which a small (sublinear) set of infected vertices creates an avalanche of infections that in turn leads to an outbreak. We show that this occurs essentially only when the tail of the weight distribution dominates a power law with exponent 3 and we determine the critical density in this case.

UR - http://www.scopus.com/inward/record.url?eid=2-s2.0-85046255319&partnerID=MN8TOARS

U2 - 10.1214/17-AAP1324

DO - 10.1214/17-AAP1324

M3 - Article

VL - 28

SP - 990

EP - 1051

JO - The annals of applied probability

JF - The annals of applied probability

SN - 1050-5164

IS - 2

ER -