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

Virtually all the emergent properties of a complex system are rooted in the non-homogeneous nature of the behaviours of its elements and of the interactions among them. However, the fact that heterogeneity and correlations can appear simultaneously at local, mesoscopic, and global scales, is a concrete challenge for any systematic approach to quantify them in systems of different types. We develop here a scalable and non-parametric framework to characterise the presence of heterogeneity and correlations in a complex system, based on the statistics of random walks over the underlying network of interactions among its units. In particular, we focus on normalised mean first passage times between meaningful pre-assigned classes of nodes, and we showcase a variety of their potential applications. We found that the proposed framework is able to characterise polarisation in voting systems, including the UK Brexit referendum and the roll-call votes in the US Congress. Moreover, the distributions of class mean first passage times can help identifying the key players responsible for the spread of a disease in a social system, and comparing the spatial segregation of US cities, revealing the central role of urban mobility in shaping the incidence of socio-economic inequalities.