Abstract
In this thesis we define a class of competing growth processes, which is a generalisationof reinforced branching processes. The class encompasses different preferential
attachment models for networks with fitness such as the Bianconi–Barab´asi tree and the network of Dereich. We analyse the asymptotic behaviour
of the largest degree of the network, which corresponds to the largest “family” of our
competing growth processes. Apart from networks, our framework also encompasses
random permutations with cycle weights (e.g. Chinese restaurant processes), and populations
with selection and mutation.
Competing growth processes can be described as a sequence of growing families,
which have different birth times and different exponential growth rates. The growth
rates are sampled from an i.i.d. sequence of bounded random variables, while the birth
times may be random and can depend on the growth process itself. In the most interesting
cases the birth times arise from an exponentially growing process so that the
largest family at time t arises in competition of the few families born early, which have
a longer time to grow, and the many families born late, among which the occurrence
of a higher birth rate is more probable.
Our main results show convergence of the scaled size of the largest family at large
times to a Fr´echet distribution and of the standardised birth time of this family to a
Gaussian distribution, in the case where the growth rates are sampled from the maximum
domain of attraction of the Gumbel distribution. Furthermore, we compare these
results to their counterparts where the growth rates are sampled from the maximum domain
of attraction of the Weibull distribution. In this case the scaled size of the largest
family also converges to a Fr´echet distribution; moreover we obtain the convergence of
the fitness of the largest family at large times to a Gamma distribution.
Date of Award  19 Feb 2020 

Original language  English 
Awarding Institution 

Supervisor  Cecile Mailler (Supervisor) & Peter Morters (Supervisor) 