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Title: Competing growth processes with applications to networks
Author: Senkevich, Anna
ISNI:       0000 0004 8506 0060
Awarding Body: University of Bath
Current Institution: University of Bath
Date of Award: 2020
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In this thesis we define a class of competing growth processes, which is a generalisation of 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 Fréchet 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 Fréchet distribution; moreover we obtain the convergence of the fitness of the largest family at large times to a Gamma distribution.
Supervisor: Mailler, Cecile ; Morters, Peter Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available