55 research outputs found
Random networks with sublinear preferential attachment: The giant component
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.Comment: Published in at http://dx.doi.org/10.1214/11-AOP697 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Emergence of condensation in Kingman's model of selection and mutation
We describe the onset of condensation in the simple model for the balance
between selection and mutation given by Kingman in terms of a scaling limit
theorem. Loosely speaking, this shows that the wave moving towards genes of
maximal fitness has the shape of a gamma distribution. We conjecture that this
wave shape is a universal phenomenon that can also be found in a variety of
more complex models, well beyond the genetics context, and provide some further
evidence for this
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