Singh, Navin (2001) *Are randomly grown graphs really random?* Physical Review E, 64 (4).

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Official URL: https://journals.aps.org/pre/abstract/10.1103/Phys...

## Abstract

We analyze a minimal model of a growing network. At each time step, a new vertex is added; then, with probability δ, two vertices are chosen uniformly at random and joined by an undirected edge. This process is repeated for t time steps. In the limit of large t, the resulting graph displays surprisingly rich characteristics. In particular, a giant component emerges in an infinite-order phase transition at δ=1/8. At the transition, the average component size jumps discontinuously but remains finite. In contrast, a static random graph with the same degree distribution exhibits a second-order phase transition at δ=1/4, and the average component size diverges there. These dramatic differences between grown and static random graphs stem from a positive correlation between the degrees of connected vertices in the grown graph—older vertices tend to have higher degree, and to link with other high-degree vertices, merely by virtue of their age. We conclude that grown graphs, however randomly they are constructed, are fundamentally different from their static random graph counterparts.

Item Type: | Article |
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Subjects: | Q Science > QC Physics |

Divisions: | Physics |

ID Code: | 1781 |

Deposited By: | Mr Mahesh Sharma |

Deposited On: | 04 Oct 2017 10:11 |

Last Modified: | 04 Oct 2017 10:11 |

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