The Annals of Probability

Random networks with sublinear preferential attachment: The giant component

Steffen Dereich and Peter Mörters

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 23 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1358951989

Digital Object Identifier
doi:10.1214/11-AOP697

Mathematical Reviews number (MathSciNet)
MR3059201

Zentralblatt MATH identifier
1260.05143

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

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

Citation

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. https://projecteuclid.org/euclid.aop/1358951989


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