First-passage times to quantify and compare structural correlations and heterogeneity in complex systems

Virtually all the emergent properties of complex systems are rooted in the non-homogeneous nature of the behaviours of their elements and of the interactions among them. However, heterogeneity and correlations appear simultaneously at multiple relevant scales, making it hard to devise a systematic approach to quantify them. We develop here a scalable and non-parametric framework to characterise the presence of heterogeneity and correlations in a complex system, based on normalised mean first passage times between preassigned classes of nodes. We showcase a variety of concrete applications, including the quantification of polarisation in the UK Brexit referendum and the roll-call votes in the US Congress, the identification of key players in disease spreading, and the comparison of spatial segregation of US cities. These results show that the diffusion structure of a system is indeed a defining aspect of the complexity of its organisation and functioning. Quantifying heterogeneity and correlations in complex systems consisting of a large number of interacting elements is key to understand their emergent properties. Here, the authors propose a method based on the statistics of interclass mean first passage times or random walks, and use it to quantify in a non-parametric way the level of heterogeneity and the presence of correlations in multiscale complex systems, where nodes are associated to a discrete number of classes.