Correlations in scale-free networks: tomography and percolation.

We discuss three related models of scale-free networks with the same degree distribution but different correlation properties. Starting from the Barabási-Albert construction based on growth and preferential attachment we discuss two other networks emerging when randomizing it with respect to links or nodes. We point out that the Barabási-Albert model displays dissortative behavior with respect to the nodes' degrees, while the node-randomized network shows assortative mixing. These kinds of correlations are visualized by discussing the shell structure of the networks around an arbitrary node. In spite of different correlation behaviors, all three constructions exhibit similar percolation properties. This result for percolation is also detected for a network with finite second moment and its corresponding randomized models.