Pearson correlations on complex networks

Complex networks are useful tools to understand propagation events like epidemics, word-of-mouth, adoption of habits and innovations. Estimating the correlation between two processes happening on the same network is therefore an important problem with a number of applications. However, at present there is no way to do so: current methods either correlate a network with itself, a single process with the network structure or calculate a network distance between two processes. In this article, we propose to extend the Pearson correlation coefficient to work on complex networks. Given two vectors, we define a function that uses the topology of the network to return a correlation coefficient. We show that our formulation is intuitive and returns the expected values in a number of scenarios. We also demonstrate how the classical the Pearson correlation coefficient is unable to do so. We conclude the article with two case studies, showcasing how our network correlation can facilitate tasks in social network analysis and economics. We provide examples of how we could use our network correlation to infer user characteristics from their activities on social media; and relationships between industrial products, under some assumptions as to what should make two exporting countries similar.