Distributions of city sizes in Mexico during the 20th century

We present a study of the distribution of cities in Mexico along the 20th century, based on information collected in censuses every ten years. The size-rank and survival cumulative distributions are constructed to evaluate the presence of scaling, its deviation from the Zipf's law and their evolution along the period of observation. We find that the size of cities S, approximately follow a power-law with the rank r,S(r)∼r-^α, where the exponents take values between α≈0.7 to α≈1.1 for years 1900 and 2000, respectively. The local fluctuations in the scaling behavior are evaluated by means of a local exponent, and the deviation of the size predicted by the Zipf's law (α=1) and the real size of each city is analyzed. Our calculations show that local exponents follow transitions between values above and below of the Zipfian regime and the deviations are more remarkable at the beginning and at the end of the 20th century. Besides, the cumulative distributions confirm the presence of scaling for the same records with a reasonable agreement with the scaling exponents observed in the size-rank distributions. Moreover, we examine the role of a recent introduced property named coherence. Finally, we explain our findings in terms of the socio-demographic evolution of Mexico along the 20th century.