  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.
