Long-range degree correlations in complex networks

Abstract

Social networks are often degree correlated between nearest neighbors, an effect termed homophily, wherein individuals connect to nearest neighbors of similar connectivity. Whether friendships or other associations are so correlated beyond the first-neighbors, and whether such correlations are an inherent property of the network or are dependent on other specifics of social interactions, remains unclear. Here we address these problems by examining long-range degree correlations in three undirected online social and three undirected nonsocial (airport, transcriptional-regulatory) networks. Degree correlations were measured using Pearson correlation scores and by calculating the average neighbor degrees for nodes separated by up to 5 sequential links. We found that the online social networks exhibited primarily weak anticorrelation at the first-neighbor level, and tended more strongly towards disassortativity as separation distances increased. In contrast, the nonsocial networks were disassortative among first-neighbors, but exhibited assortativity at longer separation distances. In addition, the average degrees of the separated neighbors approached the average network connectivity after approximately 3-4 steps. Finally, we observed that two algorithms designed to grow networks on a node-by-node basis failed to reproduce all the correlative features representative of the social networks studied here.