## Abstract

Leskovec, Kleinberg and Faloutsos (2005) observed that many social networks exhibit properties such as shrinking (i.e. bounded) diameter, densification, and (power-law) heavy tail degree distributions. To explain these phenomena, they introduced a generative model, called the Forest Fire model, and using simulations showed that this model indeed exhibited these properties; however, proving this rigorously was left as an open problem. In this paper, we analyse one of these properties, shrinking diameter. We define a restricted version of their model that incorporates the main features that seem to contribute towards this property, and prove that the graphs generated by this model exhibit shrinking distance to the seed graph. We prove that an even simpler model, the random walk model, already exhibits this phenomenon.

Original language | English |
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Title of host publication | 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 |

Editors | Robert Krauthgamer |

Publisher | Association for Computing Machinery |

Pages | 1602-1620 |

Number of pages | 19 |

ISBN (Electronic) | 9781510819672 |

DOIs | |

State | Published - 2016 |

Externally published | Yes |

Event | 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 - Arlington, United States Duration: 10 Jan 2016 → 12 Jan 2016 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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Volume | 3 |

### Conference

Conference | 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 |
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Country/Territory | United States |

City | Arlington |

Period | 10/01/16 → 12/01/16 |

### Bibliographical note

Publisher Copyright:© (2016) by SIAM: Society for Industrial and Applied Mathematics.