Universal scaling of distances in complex networks

Universal scaling of distances between vertices of Erdos-Renyi random graphs, scale-free Barabasi-Albert models, science collaboration networks, biological networks, Internet Autonomous Systems and public transport networks are observed. A mean distance distance between two nodes of degrees k_i and k_j equals to = A - B log(k_i*k_j ). The scaling is valid over several decades. A simple theory for the appearance of this scaling is presented. Parameters A and B depend on the mean value of a node degree _nn calculated for the nearest neighbors, on network clustering coefficients and degree-degree correlations. We can also explain log- periodic oscillations around this law by the discrete character of inter-node distances. References 1. Janusz A. Ho�yst, Julian Sienkiewicz, Agata Fronczak, Piotr Fronczak and Krzysztof Suchecki, Universal scaling of distances in complex networks, Phys. Rev. E 72,026108 (2005) 2. Julian Sienkiewicz, Piotr Fronczak, Janusz A. Holyst, Log-periodic oscillations due to discrete effects in complex networks, arXiv:cond-mat/0608273, Phys. Rev. E inprint, 2007.