  This paper explores the relationship between the inner economical structure
of communities and their population distribution through a rank-rank analysis
of official data, along statistical physics ideas within two techniques. The
data is taken on Italian cities. The analysis is performed both at a global
(national) and at a more local (regional) level in order to distinguish "macro"
and "micro" aspects. First, the rank-size rule is found not to be a standard
power law, as in many other studies, but a doubly decreasing power law. Next,
the Kendall and the Spearman rank correlation coefficients which measure pair
concordance and the correlation between fluctuations in two rankings,
respectively, - as a correlation function does in thermodynamics, are
calculated for finding rank correlation (if any) between demography and wealth.
Results show non only global disparities for the whole (country) set, but also
(regional) disparities, when comparing the number of cities in regions, the
number of inhabitants in cities and that in regions, as well as when comparing
the aggregated tax income of the cities and that of regions. Different outliers
are pointed out and justified. Interestingly, two classes of cities in the
country and two classes of regions in the country are found. "Common sense"
social, political, and economic considerations sustain the findings. More
importantly, the methods show that they allow to distinguish communities, very
clearly, when specific criteria are numerically sound. A specific modeling for
the findings is presented, i.e. for the doubly decreasing power law and the two
phase system, based on statistics theory, e.g., urn filling. The model ideas
can be expected to hold when similar rank relationship features are observed in
fields. It is emphasized that the analysis makes more sense than one through a
Pearson value-value correlation analysis.
