### 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. 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. Recent results of Janson, Łuczak, Turova and Vallier have shown that if the number of initially infected vertices is sublinear, then with high probability the size of the final set of infected vertices is approximately equal to a(n). That is, essentially 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 Θ(n^{1/(β - 1)}), then a_{c}(n) = n ^{β-2/β-1}.

Original language | English (US) |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Pages | 462-474 |

Number of pages | 13 |

Volume | 7695 LNCS |

DOIs | |

State | Published - 2012 |

Externally published | Yes |

Event | 8th International Workshop on Internet and Network Economics, WINE 2012 - Liverpool, United Kingdom Duration: Dec 10 2012 → Dec 12 2012 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
---|---|

Volume | 7695 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 8th International Workshop on Internet and Network Economics, WINE 2012 |
---|---|

Country | United Kingdom |

City | Liverpool |

Period | 12/10/12 → 12/12/12 |

### Fingerprint

### Keywords

- bootstrap percolation
- contagion
- power-law random graphs

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(Vol. 7695 LNCS, pp. 462-474). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7695 LNCS). https://doi.org/10.1007/978-3-642-35311-6_34

**What I tell you three times is true : Bootstrap percolation in small worlds.** / Amini, Leo Hamed; Fountoulakis, Nikolaos.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics).*vol. 7695 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 7695 LNCS, pp. 462-474, 8th International Workshop on Internet and Network Economics, WINE 2012, Liverpool, United Kingdom, 12/10/12. https://doi.org/10.1007/978-3-642-35311-6_34

}

TY - GEN

T1 - What I tell you three times is true

T2 - Bootstrap percolation in small worlds

AU - Amini, Leo Hamed

AU - Fountoulakis, Nikolaos

PY - 2012

Y1 - 2012

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. 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. Recent results of Janson, Łuczak, Turova and Vallier have shown that if the number of initially infected vertices is sublinear, then with high probability the size of the final set of infected vertices is approximately equal to a(n). That is, essentially 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 Θ(n1/(β - 1)), then ac(n) = n β-2/β-1.

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. 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. Recent results of Janson, Łuczak, Turova and Vallier have shown that if the number of initially infected vertices is sublinear, then with high probability the size of the final set of infected vertices is approximately equal to a(n). That is, essentially 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 Θ(n1/(β - 1)), then ac(n) = n β-2/β-1.

KW - bootstrap percolation

KW - contagion

KW - power-law random graphs

UR - http://www.scopus.com/inward/record.url?scp=84871385127&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84871385127&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-35311-6_34

DO - 10.1007/978-3-642-35311-6_34

M3 - Conference contribution

AN - SCOPUS:84871385127

SN - 9783642353109

VL - 7695 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 462

EP - 474

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

ER -