Unveiling Complex Territorial Socio-Economic Dynamics: A Statistical Mechanics Approach

This study proposes a novel approach based on the Ising model for analyzing socio-economic emerging patterns between municipalities by investigating the observed configuration of a network of selected territorial units which are classified as being central hubs or peripheral areas. This is interpreted as being a reference of a system of interacting territorial binary units. The socio-economic structure of the municipalities is synthesized into interpretable composite indices, which are further aggregated by means of Principal Components Analysis in order to reduce dimensionality and construct a univariate external field compatible with the Ising framework. Monte Carlo simulations via parallel computing are conducted adopting a Simulated Annealing variant of the classic Metropolis-Hastings algorithm. This ensures an efficient local exploration of the configuration space in the neighbourhood of the reference of the system. Model consistency is assessed both in terms of energy stability and the likelihood of these configurations. The comparison between observed configuration and simulated ones is crucial in the analysis of multivariate phenomena, concomitantly accounting for territorial interactions. Model uncertainty in estimating the probability of each municipality being a central hub or peripheral area is quantified by adopting the model-agnostic Conformal Prediction framework which yields adaptive intervals with guaranteed coverage. The innovative use of geographical maps of the prediction intervals renders this approach an effective tool. It combines statistical mechanics, multivariate analysis and uncertainty quantification, providing a robust and interpretable framework for modeling socio-economic territorial dynamics, with potential applications in Official Statistics.