A second spatial autocorrelation mechanism: Geographic hierarchies and urban systems

This article advances the theoretical and methodological understanding of urban hierarchies by integrating both proximity- and hierarchy-based spatial autocorrelation (SA) into spatial statistical modeling. Traditional rank-size and spatial urban systems models often fail to predict extreme city sizes, mischaracterize discrete population data as continuous, and neglect the dominant role of hierarchical SA. To address these limitations, the study summarized here introduces constrained rank-size estimations using translation parameters and direct nonlinear methods, alongside a sui-negative binomial (sui-NB) specification that properly handles over-dispersed count data. Spatial dependence modelling uses Moran eigenvector spatial filtering (MESF) with georeferenced variates conceptualized as sui- random variables, embedding SA directly into probability distributions without reliance on spatial lag terms. Empirical analysis of China, Peru, Poland, and the United States (US) urban systems shows that constrained models accurately match extreme populations—where city rank exponents can reflect primacy (Peru), missing top-tier cities (China), and structural maturity (US). MESF results reveal that hierarchical consistently outperforms contiguous SA adjustments: these former factors explain 53% of variance in the US, 60% in China, and 97% in Peru. These findings emphasize that spatial correlations frequently teleport beyond geographic proximity through tier nesting, cross-tier linkages, and within-tier lateral connections. Historical analysis further shows that global disruptions (e.g., colonialism, globalization, modernization) cause continuous positional flux in city rankings, making urban hierarchies structurally sensitive to path dependency and temporal shocks. Contradicting that rank alone explains urban systems, this paper demonstrates that a city's position in both geographic and hierarchical space also matters. As such, spatial weights matrices must be continuously updated to reflect mutating articulations, which fundamentally shape processes such as the spread of innovation or disease and, therefore, have direct implications for international, national, and regional activities.