A lot of the properties of large scale free networks are related to the power law exponent of their degree sequence. For instance, if , then supercritical networks are ultrasmall, i.e. vertex distances are typically of order , where denotes Network size. If , however typical distances are of logarithmic order (this is the socalled `small word phenomenon’). This article is concerned with the situation for and the difference behaviour in this case depending on whether there is preferential attachment in the network or not.

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