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.