A general formulation of long-range degree correlations in complex networks

We provide a general framework for analyzing degree correlations between nodes separated by more than one step (i.e., beyond nearest neighbors) in complex networks. One probability and four conditional probabilities are introduced to fully describe long-range degree correlations with respect to $k$ and $k'$ of two nodes and shortest path length $l$ between them. We present general relations among these probabilities and clarify the relevance to nearest-neighbor degree correlations. Unlike nearest-neighbor correlations, some of these probabilities are meaningful only in finite-size networks. Furthermore, as a baseline to determine the existence or nonexistence of long-range degree correlations in a network, the functional forms of these probabilities for networks without any long-range degree correlations are analytically evaluated within a mean-field approximation. The validity of our argument is demonstrated by applying it to real-world networks.