### Abstract

A bootstrap percolation process on a graph G is an "infection" process which evolves in rounds. Initially, there is a subset of infected nodes and in each subsequent round each uninfected node which has at least r infected neighbours becomes infected and remains so forever. The parameter r ≥ 2 is fixed. Such processes have been used as models for the spread of ideas or trends within a network of individuals. We analyse this process in the case where the underlying graph is an inhomogeneous random graph, which exhibits a power-law degree distribution, and initially there are a(n) randomly infected nodes. The main focus of this paper is the number of vertices that will have been infected by the end of the process. The main result of this work is that if the degree sequence of the random graph follows a power law with exponent β, where 2 <β <3, then a sublinear number of initially infected vertices is enough to spread the infection over a linear fraction of the nodes of the random graph, with high probability. More specifically, we determine explicitly a critical function a_{c}(n) such that a_{c}(n) = o(n) with the following property. Assuming that n is the number of vertices of the underlying random graph, if a(n) ≪ a_{c}(n), then the process does not evolve at all, with high probability as n grows, whereas if a(n) ≫ a_{c}(n), then there is a constant ε > 0 such that, with high probability, the final set of infected vertices has size at least εn. This behaviour is in sharp contrast with the case where the underlying graph is a G(n, p) random graph with p = d/n. It follows from an observation of Balogh and Bollobás that in this case if the number of initially infected vertices is sublinear, then there is lack of evolution of the process. It turns out that when the maximum degree is o(n^{1/(β-1)}), then a_{c}(n) depends also on r. But when the maximum degree is (Formula presented.).

Original language | English (US) |
---|---|

Pages (from-to) | 72-92 |

Number of pages | 21 |

Journal | Journal of Statistical Physics |

Volume | 155 |

Issue number | 1 |

DOIs | |

State | Published - 2014 |

Externally published | Yes |

### Fingerprint

### Keywords

- Bootstrap percolation
- Power-law random graph
- Sharp threshold

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*155*(1), 72-92. https://doi.org/10.1007/s10955-014-0946-6

**Bootstrap Percolation in Power-Law Random Graphs.** / Amini, Leo Hamed; Fountoulakis, Nikolaos.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 155, no. 1, pp. 72-92. https://doi.org/10.1007/s10955-014-0946-6

}

TY - JOUR

T1 - Bootstrap Percolation in Power-Law Random Graphs

AU - Amini, Leo Hamed

AU - Fountoulakis, Nikolaos

PY - 2014

Y1 - 2014

N2 - A bootstrap percolation process on a graph G is an "infection" process which evolves in rounds. Initially, there is a subset of infected nodes and in each subsequent round each uninfected node which has at least r infected neighbours becomes infected and remains so forever. The parameter r ≥ 2 is fixed. Such processes have been used as models for the spread of ideas or trends within a network of individuals. We analyse this process in the case where the underlying graph is an inhomogeneous random graph, which exhibits a power-law degree distribution, and initially there are a(n) randomly infected nodes. The main focus of this paper is the number of vertices that will have been infected by the end of the process. The main result of this work is that if the degree sequence of the random graph follows a power law with exponent β, where 2 <β <3, then a sublinear number of initially infected vertices is enough to spread the infection over a linear fraction of the nodes of the random graph, with high probability. More specifically, we determine explicitly a critical function ac(n) such that ac(n) = o(n) with the following property. Assuming that n is the number of vertices of the underlying random graph, if a(n) ≪ ac(n), then the process does not evolve at all, with high probability as n grows, whereas if a(n) ≫ ac(n), then there is a constant ε > 0 such that, with high probability, the final set of infected vertices has size at least εn. This behaviour is in sharp contrast with the case where the underlying graph is a G(n, p) random graph with p = d/n. It follows from an observation of Balogh and Bollobás that in this case if the number of initially infected vertices is sublinear, then there is lack of evolution of the process. It turns out that when the maximum degree is o(n1/(β-1)), then ac(n) depends also on r. But when the maximum degree is (Formula presented.).

AB - A bootstrap percolation process on a graph G is an "infection" process which evolves in rounds. Initially, there is a subset of infected nodes and in each subsequent round each uninfected node which has at least r infected neighbours becomes infected and remains so forever. The parameter r ≥ 2 is fixed. Such processes have been used as models for the spread of ideas or trends within a network of individuals. We analyse this process in the case where the underlying graph is an inhomogeneous random graph, which exhibits a power-law degree distribution, and initially there are a(n) randomly infected nodes. The main focus of this paper is the number of vertices that will have been infected by the end of the process. The main result of this work is that if the degree sequence of the random graph follows a power law with exponent β, where 2 <β <3, then a sublinear number of initially infected vertices is enough to spread the infection over a linear fraction of the nodes of the random graph, with high probability. More specifically, we determine explicitly a critical function ac(n) such that ac(n) = o(n) with the following property. Assuming that n is the number of vertices of the underlying random graph, if a(n) ≪ ac(n), then the process does not evolve at all, with high probability as n grows, whereas if a(n) ≫ ac(n), then there is a constant ε > 0 such that, with high probability, the final set of infected vertices has size at least εn. This behaviour is in sharp contrast with the case where the underlying graph is a G(n, p) random graph with p = d/n. It follows from an observation of Balogh and Bollobás that in this case if the number of initially infected vertices is sublinear, then there is lack of evolution of the process. It turns out that when the maximum degree is o(n1/(β-1)), then ac(n) depends also on r. But when the maximum degree is (Formula presented.).

KW - Bootstrap percolation

KW - Power-law random graph

KW - Sharp threshold

UR - http://www.scopus.com/inward/record.url?scp=84897028353&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84897028353&partnerID=8YFLogxK

U2 - 10.1007/s10955-014-0946-6

DO - 10.1007/s10955-014-0946-6

M3 - Article

AN - SCOPUS:84897028353

VL - 155

SP - 72

EP - 92

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1

ER -