# Fractality and degree correlations in scale-free networks

## TL;DR

This paper investigates whether disassortativity causes fractality in scale-free networks and finds that disassortativity alone does not induce fractal properties, suggesting long-range correlations are necessary.

## Contribution

The study demonstrates that disassortativity is not sufficient for fractality in scale-free networks and highlights the importance of long-range degree correlations.

## Key findings

- Most maximally disassortative networks are not fractal.
- Disassortativity does not cause fractality.
- Long-range correlations are needed for fractality.

## Abstract

Fractal scale-free networks are empirically known to exhibit disassortative degree mixing. It is, however, not obvious whether a negative degree correlation between nearest neighbor nodes makes a scale-free network fractal. Here we examine the possibility that disassortativity in complex networks is the origin of fractality. To this end, maximally disassortative (MD) networks are prepared by rewiring edges while keeping the degree sequence of an initial uncorrelated scale-free network that is guaranteed to become fractal by rewiring edges. Our results show that most of MD networks with different topologies are not fractal, which demonstrates that disassortativity does not cause the fractal property of networks. In addition, we suggest that fractality of scale-free networks requires a long-range repulsive correlation in similar degrees.