## 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 | Internet and Network Economics - 8th International Workshop, WINE 2012, Proceedings |

Pages | 462-474 |

Number of pages | 13 |

DOIs | |

State | Published - Dec 26 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) |
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Volume | 7695 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 8th International Workshop on Internet and Network Economics, WINE 2012 |
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Country | United Kingdom |

City | Liverpool |

Period | 12/10/12 → 12/12/12 |

## Keywords

- bootstrap percolation
- contagion
- power-law random graphs

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)