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Self-similar networks with scale-free degree distribution have recentlyattracted much attention, since these apparently incompatible properties werereconciled in a paper by Song et al. by an appropriate box-counting method thatenters the measurement of the fractal dimension. We study two geneticregulatory networks ({\it Saccharomyces cerevisiae} and {\it Escherichai coli}and show their self-similar and scale-free features, in extension to thedatasets studied by Song et al. Moreover, by a number of numerical results wesupport the conjecture that self-similar scale-free networks are notassortative. From our simulations so far these networks seem to bedisassortative instead. We also find that the qualitative feature ofdisassortativity is scale-invariant under renormalization, but it appears as anintrinsic feature of the renormalization prescription, as even assortativenetworks become disassortative after a sufficient number of renormalizationsteps.Comment: 4 pages, 6 figure

Self-similar scale-free networks and disassortativity