Self-similar Scale-free Networks and Disassortativity

TL;DR

This paper investigates self-similar, scale-free networks, including biological networks, and finds that they tend to be disassortative, with this property being scale-invariant and intrinsic to the renormalization process.

Contribution

It extends the study of self-similar scale-free networks to biological systems and reveals their disassortative nature through numerical analysis and renormalization.

Findings

Biological networks exhibit self-similarity and scale-free properties.

Self-similar scale-free networks tend to be disassortative.

Disassortativity is scale-invariant and intrinsic to renormalization.

Abstract

Self-similar networks with scale-free degree distribution have recently attracted much attention, since these apparently incompatible properties were reconciled in a paper by Song et al. by an appropriate box-counting method that enters the measurement of the fractal dimension. We study two genetic regulatory networks ({\it Saccharomyces cerevisiae} and {\it Escherichai coli} and show their self-similar and scale-free features, in extension to the datasets studied by Song et al. Moreover, by a number of numerical results we support the conjecture that self-similar scale-free networks are not assortative. From our simulations so far these networks seem to be disassortative instead. We also find that the qualitative feature of disassortativity is scale-invariant under renormalization, but it appears as an intrinsic feature of the renormalization prescription, as even assortative networks become disassortative after a sufficient number of renormalization steps.