  We propose a model of a growing network, in which preferential linking is
combined with partial inheritance of connectivity (number of incoming links) of
individual nodes by new ones. The nontrivial version of this model is solved
exactly in the limit of a large network size. We demonstrate, that the
connectivity distribution depends on the network size, $t$, in a {\em
multifractal} fashion. When the size of the network tends to infinity, the
distribution behaves as $\sim q^{-\gamma}\ln q$, where $\gamma =\sqrt{2}$. For
the finite-size network, this behavior is observed for $1 \ll q \lesssim
\exp(\ln ^{1/2}t) $ but the multifractality is determined by the far wider
part, $1 \ll q \lesssim \sqrt t$, of the distribution function.
