Radiation model for migration with directional preferences

The radiation model is a parameter-free model of human mobility that has been applied primarily for short-distance moves, such as commuting. When applied to migration, it underestimates the number of long-range moves, such as between different US states. Here we show that it additionally suffers from a conceptual inconsistency that can have substantial numerical effects on long-distance moves. We propose a modification of the radiation model that introduces a dependence on the angle between any two alternative potential destinations, accounting for the possibility that migrants may have preferences about the approximate direction of their move. We demonstrate that this modification mitigates the conceptual inconsistency and improves the model fit to observational migration data, without introducing any fitting parameters.

Figure 1

(a) Migration scenario, with origin $i$ and destinations $k$ and $j$. Here $d_{ik}$ and $d_{ij}$ denote the respective distances, $s_{ik}$ denotes the intervening opportunities for a migration flow between $i$ and $k$, and ${\alpha }_{kj}$ is the angle between $k$ and $j$. (b) and (c) Two migration scenarios originating from Kings County with the destination Los Angeles or Santa Clara. Each partial circle represents the radius indicating which population counts in the intervening opportunities and which do not. (b) The distance between KC and SC is 210 km and between KC and LA it is 252 km, making SC part of LA's intervening opportunities. (c) Swapped distances. The distance between KC and SC is now 252 km and that between KC and LA is 210 km, making LA part of SC's intervening opportunities.

Figure 2

Population count and $s_{ij}$ for counties within a circle (excluding areas outside the USA) around Salt Lake City (the radius is 1100 km and the ring width 240 km) (left column), Kansas City (radius of 875 km and ring width of 210 km) (middle column), and Minneapolis (radius of 825 km and ring width of 210 km) (right column). The top row displays the destination population, the second row the $s_{ij}$ of the original RM for each county in the outer circle being treated as a potential destination, the third row the optimal $s_{ij}$, calculated with Eq. (12), and the bottom row the $s_{ij}$ of the ADRM [Eq. (7)].

Figure 3

Migration rate ratios for city triplets, consisting of one origin $i$ and two destinations $k$ and $j$ in relation to the angle ${\alpha }_{kj}$. The distances fulfill the condition $0.6d_{ij}<d_{ik}<0.9d_{ij}$ as well as (a) the origin population less than or equal to 170 000, destination population greater than or equal to 1 000 000, and distance greater than 100 and less than 1300 and (b) origin population greater than or equal to 1 000 000, destination population greater than or equal to 1 000 000, and distance greater than 100 and less than 2000. The sample size is (a) 1094 and (b) 1140. Each panel shows the data binned into multiples of $\pi /10$. Boxes indicate the 25th and 75th percentiles, the whiskers indicate the 5th and 95th percentiles, the orange lines indicate the median values for each bin, black lines indicate the linear fit through the median values, and gray lines indicate a ratio of one.

Figure 4

Performance of the ADRM and RM tested on US internal migration data. The left column shows the difference between the Sørensen-Dice coefficient for the ADRM and RM depending on the origin population and distance (top) and destination population and distance (bottom). The middle column shows the number of migrants depending on the origin population and distance (top) and destination population and distance (bottom). The right column shows the difference between the $R^2$ score for the ADRM and RM depending on the origin population and distance (top) and destination population and distance (bottom). Each square indicates the difference between the performance measure of the original RM and the ADRM, red indicates that the ADRM performs better and blue that the RM performs better. The middle column is shown to indicate which population and distance regimes are most frequented. White lines in the plot indicate no sample data for these population and distance combinations.

Figure 5

Performance of the ADRM and RM tested on Mexican internal migration data. The left column shows the difference between the Sørensen-Dice coefficient for the ADRM and RM depending on the origin population and distance (top) and destination population and distance (bottom). The middle column shows the number of migrants depending on the origin population and distance (top) and destination population and distance (bottom). The right column shows the difference between the $R^2$ score for the ADRM and RM depending on the origin population and distance (top) and destination population and distance (bottom). Each square indicates the difference between the performance measure of the original RM and the ADRM, red indicates that the ADRM performs better and blue that the RM performs better. The middle column is shown to indicate which population and distance regimes are most frequented.

Figure 6

Performance of the original radiation model and ADRM tested on US internal migration data, dependent on traveled distance (top), origin population size (middle), and destination population size (bottom). The left column shows the Sørensen-Dice (SD) coefficient and the right column the $R^2$ score. Each bar denotes the performance value for an interval of 300 km or $200000$ persons, respectively. The $R^2$ is clipped between $-0.5$ and 1.

Figure 7

Performance of the RM and ADRM on Mexican internal migration data. The left column shows the Sørensen-Dice coefficient and the right the $R^2$ score. The top row shows the performance depending on the traveled distance, the middle row depending on the origin population, and the bottom row depending on the destination population. Each marker denotes the performance value for an interval of 100 km or $120000$ persons, respectively. The $R^2$ is clipped between $-1$ and 1.