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2009 Random networks with sublinear preferential attachment: Degree evolutions
Steffen Dereich, Peter Mörters
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Electron. J. Probab. 14: 1222-1267 (2009). DOI: 10.1214/EJP.v14-647


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.


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Steffen Dereich. Peter Mörters. "Random networks with sublinear preferential attachment: Degree evolutions." Electron. J. Probab. 14 1222 - 1267, 2009.


Accepted: 3 June 2009; Published: 2009
First available in Project Euclid: 1 June 2016

zbMATH: 1185.05127
MathSciNet: MR2511283
Digital Object Identifier: 10.1214/EJP.v14-647

Primary: 05C80
Secondary: 60C05 , 90B15

Keywords: Barabasi-Albert model , degree distribution , Dynamic random graphs , large deviation principle , maximal degree , Moderate deviation principle , sublinear preferential attachment

Vol.14 • 2009
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