  We study the following paradox associated with networks growing according to
superlinear preferential attachment: superlinear preference cannot produce
scale-free networks in the thermodynamic limit, but there are superlinearly
growing network models that perfectly match the structure of some real
scale-free networks, such as the Internet. We obtain an analytic solution,
supported by extensive simulations, for the degree distribution in
superlinearly growing networks with arbitrary average degree, and confirm that
in the true thermodynamic limit these networks are indeed degenerate, i.e.,
almost all nodes have low degrees. We then show that superlinear growth has
vast preasymptotic regimes whose depths depend both on the average degree in
the network and on how superlinear the preference kernel is. We demonstrate
that a superlinearly growing network model can reproduce, in its preasymptotic
regime, the structure of a real network, if the model captures some
sufficiently strong structural constraints -- rich-club connectivity, for
example. These findings suggest that real scale-free networks of finite size
may exist in preasymptotic regimes of network evolution processes that lead to
degenerate network formations in the thermodynamic limit.
