The Annals of Probability

Random networks with sublinear preferential attachment: The giant component

Steffen Dereich and Peter Mörters

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We study a dynamical random network model in which at every construction step a new vertex is introduced and attached to every existing vertex independently with a probability proportional to a concave function $f$ of its current degree. We give a criterion for the existence of a giant component, which is both necessary and sufficient, and which becomes explicit when $f$ is linear. Otherwise it allows the derivation of explicit necessary and sufficient conditions, which are often fairly close. We give an explicit criterion to decide whether the giant component is robust under random removal of edges. We also determine asymptotically the size of the giant component and the empirical distribution of component sizes in terms of the survival probability and size distribution of a multitype branching random walk associated with $f$.

Article information

Ann. Probab., Volume 41, Number 1 (2013), 329-384.

First available in Project Euclid: 23 January 2013

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Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20]
Secondary: 60C05: Combinatorial probability 90B15: Network models, stochastic

Barabási–Albert model Erdős–Rényi model power law scale-free network nonlinear preferential attachment dynamic random graph giant component cluster multitype branching random walk


Dereich, Steffen; Mörters, Peter. Random networks with sublinear preferential attachment: The giant component. Ann. Probab. 41 (2013), no. 1, 329--384. doi:10.1214/11-AOP697.

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