  Connectivity correlations play an important role in the structure of
scale-free networks. While several empirical studies exist, there is no general
theoretical analysis that can explain the largely varying behavior of real
networks. Here, we use scaling theory to quantify the degree of correlations in
the particular case of networks with a power-law degree distribution. These
networks are classified in terms of their correlation properties, revealing
additional information on their structure. For instance, the studied social
networks and the Internet at the router level are clustered around the line of
random networks, implying a strongly connected core of hubs. On the contrary,
some biological networks and the WWW exhibit strong anti-correlations. The
present approach can be used to study robustness or diffusion, where we find
that anti-correlations tend to accelerate the diffusion process.
