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

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