# Random networks with sublinear preferential attachment:degree evolutions

### Reference:

Dereich, S. and Morters, P., 2009. Random networks with sublinear preferential attachment:degree evolutions. *Electronic Journal of Probability*, 14, pp. 1222-1267.

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### Official URL:

http://dx.doi.org/10.1214/EJP.v14-647

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### 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.

### Details

Item Type | Articles | ||||||

Creators | Dereich, S.and Morters, P. | ||||||

DOI | 10.1214/EJP.v14-647 | ||||||

Related URLs |
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Uncontrolled Keywords | dynamic random graphs,degree distribution,maximal degree,sublinear preferential attachment,moderate deviation principle,large deviation principle,barabasi-albert model | ||||||

Departments | Faculty of Science > Mathematical Sciences | ||||||

Research Centres | Centre for Mathematical Biology | ||||||

Refereed | Yes | ||||||

Status | Published | ||||||

ID Code | 14716 | ||||||

Additional Information | ID number: Article Number 43 |

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