Preferential attachment with choice

We consider the degree distributions of preferential attachment ran-dom graph models with choice similar to those considered in recent work by Malyshkin and Paquette and Krapivsky and Redner. In these models a new vertex chooses r vertices according to a preferential rule and connects to the vertex in the selection with the sth highest degree. For meek choice, where s> 1, we show that both double exponential decay of the degree distribution and condensation-like behaviour are possible, and provide a criterion to distinguish between them. For greedy choice, where s = 1, we confirm that the degree distribution asympotically follows a power law with logarithmic correction when r = 2 and shows condensation-like be-haviour when r> 2. AMS 2010 Subject Classification: Primary 05C82. Key words and phrases:random graphs; preferential attachment; choice.