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A branching process with fitness

Speaker: Igor Smolyarenko (Brunel)

Abstract

The model under consideration is the reinforced branching process. Variants of this model include the Bianconi-Barabasi (BB) model of preferential attachment with fitness. The model describes a growing population of individuals (e.g., bacteria, or half-edges of a network), each characterised by a fitness parameter which controls its reproduction rate. The off-spring may either inherit the parent's fitness (i.e., grow the family) or acquire a mutated fitness value drawn from a given distribution (i.e., found a new family). The model exhibits emergence of a small number of dominant families, akin to Bose-Einstein condensation. A subject of some controversy, it has been recently established that, contrary to the original "winner-take-all" conjecture of BB, for a somewhat restricted class of fitness distributions with finite support no single family contains a finite fraction of the overall population. I will analyse the population dynamics and family size distribution inside the "condensate"for arbitrary fitness distributions.  The results in the infinite support case are consistent with the existence of "winner-take-all" families.