Typical distances in ultrasmall random networks

We show that in preferential attachment models with power-law exponent $τ\in(2,3)$ the distance between randomly chosen vertices in the giant component is asymptotically equal to $(4+o(1))\, \frac{\log\log N}{-\log (τ-2)}$, where $N$ denotes the number of nodes. This is twice the value obtained for several types of configuration models with the same power-law exponent. The extra factor reveals the different structure of typical shortest paths in preferential attachment graphs.

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