## 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, Redner and Leyvraz (2000) on how the condensation arises with respect to the degree distribution.

60G20, O5C20, 37H10

Original language | English (US) |
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Journal | Unknown Journal |

State | Published - Apr 18 2017 |

## Keywords

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

## ASJC Scopus subject areas

- General