Scaling of distances in correlated complex networks

The influence of node–node degree correlations on distances in complex networks has been studied. We have found that even the presence of strong correlations in complex networks does not break a universal scaling of distances between vertices of such networks as science collaboration networks, biological networks, Internet Autonomous Systems and public transport systems. A mean distance between two nodes of degrees ki and kj in such networks equals to 〈lij〉=A-Blog(kikj) for a fixed value of the product kikj. The scaling is valid over several decades. Parameters A and B depend on the mean value of a node degree 〈k〉nn calculated for the nearest neighbors. We have found that extending our simple theory basing on a random branching tree by the first-order node degree correlations improves theoretical predictions for parameters A and B in assortative networks, while it fails in disassortative ones.