### 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) |
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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 |
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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 |
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Country | Germany |

City | Berlin |

Period | 8/15/16 → 8/19/16 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### 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

**On the Growth of a Superlinear Preferential Attachment Scheme.** / Sethuraman, Sunder; Venkataramani, Shankar C.

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

*Probability and Analysis in Interacting Physical Systems - In Honor of S.R.S. Varadhan, 2016.*Springer Proceedings in Mathematics and Statistics, vol. 283, Springer New York LLC, pp. 243-265, Conference in Honor of the 75th Birthday of S.R.S. Varadhan, 2016, Berlin, Germany, 8/15/16. https://doi.org/10.1007/978-3-030-15338-0_9

}

TY - GEN

T1 - On the Growth of a Superlinear Preferential Attachment Scheme

AU - Sethuraman, Sunder

AU - Venkataramani, Shankar C

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - Degree distribution

KW - Fluctuations

KW - Growth

KW - Preferential attachment

KW - Random graphs

KW - Superlinear

UR - http://www.scopus.com/inward/record.url?scp=85068959028&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85068959028&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-15338-0_9

DO - 10.1007/978-3-030-15338-0_9

M3 - Conference contribution

AN - SCOPUS:85068959028

SN - 9783030153373

T3 - Springer Proceedings in Mathematics and Statistics

SP - 243

EP - 265

BT - Probability and Analysis in Interacting Physical Systems - In Honor of S.R.S. Varadhan, 2016

A2 - Olla, Stefano

A2 - Friz, Peter

A2 - König, Wolfgang

A2 - Mukherjee, Chiranjib

PB - Springer New York LLC

ER -