### Abstract

We consider an evolving preferential attachment random graph model where at discrete times a new node is attached to an old node, selected with probability proportional to a superlinear function of its degree. For such schemes, it is known that the graph evolution condenses, that is a.s. in the limit graph there will be a single random node with infinite degree, while all others have finite degree. In this note, we establish a.s. law of large numbers type limits and fluctuation results, as n ↑ ∞, for the counts of the number of nodes with degree k ≥ 1 at time n ≥ 1. These limits rigorously verify and extend a physical picture of Krapivisky et al. (Phys Rev Lett 85:4629–4632, 2000 [16]) on how the condensation arises with respect to the degree distribution.

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

Title of host publication | Probability and Analysis in Interacting Physical Systems - In Honor of S.R.S. Varadhan, 2016 |

Editors | Stefano Olla, Peter Friz, Wolfgang König, Chiranjib Mukherjee |

Publisher | Springer New York LLC |

Pages | 243-265 |

Number of pages | 23 |

ISBN (Print) | 9783030153373 |

DOIs | |

State | Published - Jan 1 2019 |

Event | Conference in Honor of the 75th Birthday of S.R.S. Varadhan, 2016 - Berlin, Germany Duration: Aug 15 2016 → Aug 19 2016 |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
---|---|

Volume | 283 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Conference

Conference | Conference in Honor of the 75th Birthday of S.R.S. Varadhan, 2016 |
---|---|

Country | Germany |

City | Berlin |

Period | 8/15/16 → 8/19/16 |

### Keywords

- Degree distribution
- Fluctuations
- Growth
- Preferential attachment
- Random graphs
- Superlinear

### ASJC Scopus subject areas

- Mathematics(all)

## Fingerprint Dive into the research topics of 'On the Growth of a Superlinear Preferential Attachment Scheme'. Together they form a unique fingerprint.

## Cite this

*Probability and Analysis in Interacting Physical Systems - In Honor of S.R.S. Varadhan, 2016*(pp. 243-265). (Springer Proceedings in Mathematics and Statistics; Vol. 283). Springer New York LLC. https://doi.org/10.1007/978-3-030-15338-0_9