Lower bound of assortativity coefficient in scale-free networks

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 $p(k)\sim k^{-{\hat{I}}^3}$, 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 ${\hat{I}}^3$. 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.

9 pages, 5 figures and 1 Table