Assortativity Properties of Scale-Free Networks

Nodes distribution by degrees is the most important characteristic of complex networks, but not comprehensive one. While degree distribution is a first order graph metric, the assortativity is a second order one. Assortativity coefficient is a measure of a tendency for nodes in networks to connect with similar or dissimilar ones in some way. As a simplest case, assortative mixing is considered according to nodes degree. In general, degree distribution forms an essential restriction both on the network structure and on assortativity coefficient boundaries. The problem of determining the structure of SF- networks having an extreme assortativity coefficient is considered. The estimates of boundaries for assortativity coefficient have been found. It was found, that these boundaries are as wider as scaling factor of SF-model is far from one of BA-model. In addition, the boundaries are narrowing with increasing the network size.