Preferential attachment with choice.

*(English)*Zbl 1341.05230Summary: We consider the degree distributions of preferential attachment random graph models with choice similar to those considered in recent work by Y. Malyshkin and E. Paquette [Electron. Commun. Probab. 19, Paper No. 44, 13 p. (2014; Zbl 1298.05287)] and P. L. Krapivsky and S. Redner [“Choise-driven phase transition in complex networks”, Preprint, arXiv:1312.7803]. In these models a new vertex chooses \(r\) vertices according to a preferential rule and connects to the vertex in the selection with the \(s\)th 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 asymptotically follows a power law with logarithmic correction when \(r=2\) and shows condensation-like behaviour when \(r>2\).

##### MSC:

05C80 | Random graphs (graph-theoretic aspects) |

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