Metropolitan growth in Europe has resulted in drastic changes of urban forms, socio-spatial structures and land-use patterns due to sequential processes of urbanization, suburbanization and re-urbanization. To assess latent shifts from mono-centric models towards more disarticulated and decentralized settlement configurations, the present study evaluates spatio-temporal patterns of growth between the 1920s and the 2010s in three Mediterranean cities with different structure and functions (Barcelona: compact and moderately polycentric; Rome: dispersed, medium-density; Athens: mono-centric, hyper-compact). To identify and characterize long-term urban transformations, an original approach was illustrated in this study, based on a multivariate analysis of 13 indicators resulting from descriptive statistics and linear regression modeling the relationship between population density and distance from inner cities. The empirical results of this study indicate that Barcelona, Rome and Athens have experienced different urbanization cycles, characterized by a (more or less) concentrated distribution of population along urban gradients. Despite similarities in demographic dynamics and planning practices, these processes have determined (i) a mostly centralized growth in Barcelona, (ii) a relatively dispersed and discontinuous spatial structure in Rome, and (iii) a steep decline of population density with the distance from downtown Athens. Compact urban expansion, population decline and urban de-concentration were finally assessed using the analytical approach proposed in this study.

In this paper, we introduce a penalized version of the geometric mean. In analogy with the Mazziotta Pareto Index, this composite indicator is derived as a product between the geometric mean and a penalization term to account for the unbalance among indicators. The unbalance is measured in terms of the (horizontal) variability of the normalized indicators opportunely scaled and transformed via the Box–Cox function of order zero. The penalized geometric mean is used to compute the penalized Human Development Index (HDI), and a comparison with the geometric mean approach is presented. Data come from the Human Development Data Center for 2019 and refer to the classical three dimensions of HDI. The results show that the new method does not upset the original ranking produced by the HDI but it impacts more on countries with poor performances. The paper has the merit of proposing a new reading of the Mazziotta Pareto Index in terms of the reliability of the arithmetic mean as well as of generalizing this reading to the geometric mean approach.