Lower bound of assortativity coefficient in scale-free networks

The degree-degree correlation is important in understanding the structural organization of a network and dynamics upon a network. Such correlation is usually measured by the assortativity coefficient r, with natural bounds r∈[-1,1]. For scale-free networks with power-law degree distribution p(k)∼k^-γ, 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 γ. This work challenges the validation of the 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 alone.