  We propose a preferential attachment model for network growth where new
entering nodes have a partial information about the state of the network. Our
main result is that the presence of bounded information modifies the degree
distribution by introducing an exponential tail, while it preserves a power law
behaviour over a finite small range of degrees. On the other hand, unbounded
information is sufficient to let the network grow as in the standard
Barab\'asi-Albert model. Surprisingly, the latter feature holds true also when
the fraction of known nodes goes asymptotically to zero. Analytical results are
compared to direct simulations.
