Lower bound of assortativity coefficient in scale-free networks

TL;DR

This paper analytically determines the lower bound of the assortativity coefficient in large scale-free networks, revealing it depends on the degree distribution exponent and challenging previous assumptions about network correlations.

Contribution

It provides the first analytical expression for the lower bound of assortativity in scale-free networks, highlighting limitations in interpreting correlation signs.

Findings

Lower bound depends on power-law exponent γ

Assortativity bounds are not always -1 or 1 in scale-free networks

Implications for assessing network correlations

Abstract

The degree-degree correlation is important in understanding the structural organization of a network and the dynamics upon a network. Such correlation is usually measured by the assortativity coefficient $r$, with natural bounds $r \in [-1,1]$. For scale-free networks with power-law degree distribution $p(k) \sim k^{-\gamma}$, we analytically obtain the lower bound of assortativity coefficient in the limit of large network size, which is not -1 but dependent on the power-law exponent $\gamma$. This work challenges the validation of assortativity coefficient in heterogeneous networks, suggesting that one cannot judge whether a network is positively or negatively correlated just by looking at its assortativity coefficient.