  A network growth mechanism based on a two-step preferential rule is
investigated as a model of network growth in which no global knowledge of the
network is required. In the first filtering step a subset of fixed size $m$ of
existing nodes is randomly chosen. In the second step the preferential rule of
attachment is applied to the chosen subset. The characteristics of thus formed
networks are explored using two approaches: computer simulations of network
growth and a theoretical description based on a master equation. The results of
the two approaches are in excellent agreement. Special emphasis is put on the
investigation of the node degree probability distribution. It is found that the
tail of the distribution has the exponential form given by $exp(-k/m)$.
Implications of the node degree distribution with such tail characteristics are
briefly discussed.
