Fig 1.
Selected scaling relationships fitted to the data.
The dots represent data on each of the 385 cities in the US with its size on the horizontal axis and its corresponding figures for migration given on the vertical axis given in different units (as a probability or as the inflow of migrants per 1,000 inhabitants). Also plotted on the same diagrams are the results of the scaling relationship fitted to the data with the coefficient given in each case. Top panel: three sublinear relationships. Bottom Panel: three superlinear relationships. A coefficient
, as establish in the diagram on the left in the bottom panel (for the inflow from a city larger than 5 million), means that city size has negligible impact on that flux. This establishes the approximate city size where a phase transition occurs.
Fig 2.
Probability of migration conditional on city size.
Probability of migrating from a city of a given size (horizontal axis) to a city of a given size (vertical axis). The fitted values of according to the city size (plotted in the lower panel) indicates if the probability of migrating to a destination with a given size follows a sublinear (
, in blue) or superlinear (
, in orange) behaviour. For example, we observe that if the city of origin is larger than 4 million inhabitants, then the probability of migration follows a superlinear behaviour. In contrast, a strong sublinear behaviour is observed for small cities, particularly if the city has less than 100,000 inhabitants.
Fig 3.
Schematic model of migration dynamics considering the location of an individual between two consecutive years.
An individual from the countryside decides to migrate to a city from one year to the next one (with probability ) and the destination is chosen following a sublinear pattern. An individual from a city might migrate to the countryside (with a probability that decreases sublinearly with city size) or might decide to move to another city (with a probability that also decreases sublinearly with city size) although in this case, the destination is selected according to the city size of the origin and destination (Fig 2). Finally, an individual who arrives from another country picks their destination following a superlinear pattern according to city size.
Table 1.
Results of the scaling, gravity, radiation and gravity-scaling models.
Mean square error and maximum error comparing the migration flow considering all pairs of cities as origin and destination. The smallest mean square error and the smallest maximum square error (in absolute value) are provided by the gravity-scaling model.
Fig 4.
Observed outflow and inflow of migrants from each city against the predicted values of the scaling and the gravity model.
The horizontal axis is the observed outflow or inflow of migrants from each city and the vertical axis is the results of the scaling and gravity models. The yellow line represents the identity (where the predicted value of the outflow or inflow of migrants from each city match the observed values, so there is a perfect match), so that observations closer to that line have a better fit. The gravity model (with blue colour) shows a systematic bias on the smaller cities.
Table 2.
Coefficients obtained for the outflow of internal migration in the US.
Table 3.
Coefficients obtained for the inflow of migration to the cities and the countryside in the US.