Growing network with heritable connectivity of nodes
  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. 
S.N. Dorogovtsev, J.F.F. Mendes, and A.N. Samukhin (University of   Porto and Ioffe Institute)