Projects per year

### Abstract

We define a dynamic model of random networks, where new vertices are connected to old ones with a probability proportional to a sublinear function of their degree. We first give a strong limit law for the empirical degree distribution, and then have a closer look at the temporal evolution of the degrees of individual vertices, which we describe in terms of large and moderate deviation principles. Using these results, we expose an interesting phase transition: in cases of strong preference of large degrees, eventually a single vertex emerges forever as vertex of maximal degree, whereas in cases of weak preference, the vertex of maximal degree is changing infinitely often. Loosely speaking, the transition between the two phases occurs in the case when a new edge is attached to an existing vertex with a probability proportional to the root of its current degree.

Original language | English |
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Pages (from-to) | 1222-1267 |

Journal | Electronic Journal of Probability |

Volume | 14 |

DOIs | |

Publication status | Published - 3 Jun 2009 |

### Keywords

- dynamic random graphs
- degree distribution
- maximal degree
- sublinear preferential attachment
- moderate deviation principle
- large deviation principle
- Barabasi-Albert model

## Fingerprint Dive into the research topics of 'Random networks with sublinear preferential attachment: degree evolutions'. Together they form a unique fingerprint.

## Projects

- 1 Finished

### INTERSECTION LOCAL TIMES AND STOCHASTIC PROCESSES IN RANDOM MEDIA

Morters, P.

Engineering and Physical Sciences Research Council

1/09/05 → 31/08/10

Project: Research council

## Cite this

Dereich, S., & Morters, P. (2009). Random networks with sublinear preferential attachment: degree evolutions.

*Electronic Journal of Probability*,*14*, 1222-1267. https://doi.org/10.1214/EJP.v14-647