Abstract
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.
Highlights
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
The method we propose moves from the classical research on mean first passage times (MFPT) between pairs of nodes in a graph[9,10,11,12], and focuses instead on the distribution of class mean first passage times (CMFPT), i.e., the expected number of steps ταβ needed to a random walker to visit for the first time a node of a certain class β when it starts from a node of class α
We have shown here that the information captured by the distribution of inter-class MFPTs can be used not just as a way to detect the presence of anisotropy and correlations in the properties of nodes, and as a reliable proxy for the dynamics and emergent behaviours of a complex system
Summary
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. 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 a defining aspect of the complexity of its organisation and functioning. We test our framework on a variety of systems with simple geometries and ad-hoc class assignments, and we use it in three real-world scenarios, namely the quantification of polarisation in the Brexit referendum and in the US Congress since 1926, the role of face-to-face interactions among individuals in the spread of an epidemics and the relation between economic segregation and prevalence of crime in the 53 US cities with more than one million citizens
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