Asymptotic properties of degree-correlated scale-free networks

The possible correlation profiles of networks with a given scale-free degree distribution are restricted and bounded by maximally correlated configurations. Dissortative networks consist of nested bilayers, in which low-degree vertices are connected to high-degree vertices. The number of these bilayers attains a constant value for large network size N. Assortative networks exhibit monolayers of low-degree vertices, the number of which grows monotonously with N. Analytical relations for the Pearson correlation coefficient r of these extremal configurations are derived and shown to provide lower and upper bounds on the possible r values. Both bounds are found to vanish for large networks.