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.

First passage times for Slepian process with linear and piecewise linear barriers

In this paper, we derive explicit formulas for the first-passage probabilities of the process S(t) = W(t) − W(t + 1), where W(t) is the Brownian motion, for linear and piece-wise linear barriers on arbitrary intervals [0,T]. Previously, explicit formulas for the first-passage probabilities of this process were known only for the cases of a constant barrier or T ≤ 1. The first-passage probabilities results are used to derive explicit formulas for the power of a familiar test for change-point detection in the Wiener process.