On the existence of Scaling laws across Indian districts: A new prospect for urban scaling

Urban scaling analysis in generally performed based on cities. In this study, we have (empirically) explored a new prospect for urban scaling analysis based on relatively larger local administrative units, which are independently functional, within a country. For this purpose, we have studied the scaling laws across Indian districts for the various socio-economic indicators (SEIs) within four distinct urbanization classes, namely rural, semi-rural, semi-urban and urban districts, and presented the estimated values of the scaling factors for each classes for the years 2001 and 2011 along with their goodness-of-fit measured by the $R^2$ values. Our result shows that the scaling laws indeed exist even at the district level for all most of the SEIs considered, related to education, employment, housing, health, etc.; the $R^2$ values obtained for these SEIs are very high (often greater than 0.8 or 0.9) in both the the years. For a few SEIs, however, the validity of scaling law increases as we move from rural to urban class of districts. The linearity of the scaling index has been statistically tested and it has been found, at 95\% level of confidence, that not all the SEIs behave linearly; some of them are characterized by super-linear behaviour and some behave sub-linearly. Statistical hypothesis tests have also been performed to test the equality of two scaling factors corresponding to two distinct classes and two different years to understand the differences in scaling relationships among increasing urbanisation classes and their changes over time.