Six stylized cities, sized by population, exchange animated migrant flow particles according to the gravity model of migration: bigger cities and closer pairs generate denser, more frequent flows along the connecting line.
The gravity model of migration proposes that the flow of migrants between two places is proportional to the product of their populations and inversely proportional to the distance between them raised to a decay exponent gamma — directly analogous to Newton's law of gravitation.
Drag the distance-decay slider gamma to control how sharply flows fall off with distance. Adjust the Northgate and Southhaven population sliders and watch flow lines and particle density respond instantly. Toggle labels to see numeric M_ij values, or restrict the view to the top 5 strongest routes.
The gravity model was first applied to human migration by sociologist Ernst Georg Ravenstein in the 1880s and later formalized mathematically in the 1940s-1960s. It still underlies modern census bureau and World Bank migration-flow estimates today.
This simulation visualizes the gravity model of migration, one of the oldest and most widely used tools in spatial demography for predicting how many people move between two places. The core formula, M_ij = k · Pop_i · Pop_j / dist_ij^gamma, states that migration flow between region i and region j grows with the product of their populations (bigger places generate and attract more movers) and shrinks with the distance between them raised to a decay exponent gamma (farther places exchange fewer migrants). Six stylized cities are placed on a map, each sized by population, and animated particles travel along the connecting lines at a rate proportional to the computed flow — denser particle streams mean stronger predicted migration between that pair of cities.
The gravity model traces its intellectual roots to Ernst Georg Ravenstein's 1885 "Laws of Migration," which first proposed that migration volume relates to distance and city size, echoing physical gravitation. It was mathematically formalized in the mid-20th century by researchers such as George Zipf and later refined into the modern spatial-interaction framework used by geographers and economists. Today, variants of the gravity model underpin official migration estimates from national statistics offices, the World Bank, and the United Nations, as well as models of trade flows, commuting patterns, and telecommunications traffic — anywhere two connected populations exchange volume that depends on size and separation.
The gravity model of migration predicts the volume of migrants moving between two places using a formula directly borrowed from Newton's law of gravitation: flow is proportional to the product of the two populations and inversely proportional to the distance between them raised to some power. Larger, closer cities exchange more migrants; smaller, farther cities exchange fewer. It is a simple but surprisingly accurate first approximation used throughout demography, economics, and geography.
Six cities appear on the map, each sized according to its population. Animated dots travel between city pairs, with denser streams of particles along routes that carry higher computed flow. Use the gamma slider to change how quickly flow decays with distance, and the two population sliders to grow or shrink Northgate and Southhaven — watch the flow lines and particle density update immediately. Toggle labels to see exact M_ij values, or restrict the view to only the five strongest routes.
Gamma controls how sharply migration flow falls off as distance increases. At gamma = 1, flow decreases linearly with distance (a gentle decay, similar to some short-range commuting models). At gamma = 2, the classic "inverse-square law" value borrowed directly from physics, flow drops off much faster. Higher gamma values (up to 3 in this simulation) represent settings where distance is a very strong deterrent to migration, common in models of long-distance international migration where travel cost and unfamiliarity weigh heavily.
The formula M_ij = k · Pop_i · Pop_j / dist_ij^gamma is a direct spatial-interaction analogue of Newton's law of universal gravitation, F = G · m1 · m2 / r^2, where population substitutes for mass and a tunable exponent gamma replaces the fixed exponent 2. The constant k is a calibration parameter fitted to observed migration data, absorbing factors like currency, time units, and unmeasured push-pull forces. Researchers estimate k and gamma empirically by fitting the model (often in log-linear form) to historical migration matrices using regression, which is how national statistical agencies calibrate their own migration projections.
The basic two-variable gravity model explains a substantial share of variation in real migration flows, often 60-80% of variance in log-flow regressions, which is remarkable given its simplicity. However, it systematically misses factors such as language and cultural similarity, historical colonial ties, immigration policy, employment opportunity differentials, and network effects where migrants follow established diaspora communities. Modern applied versions add these as extra multiplicative terms, producing "augmented gravity models" that can explain well over 90% of variance in flows between countries.
Because the population term enters as a product (Pop_i × Pop_j), a pair involving one very large city will tend to generate a bigger raw flow number than two small cities, even if the small cities happen to be closer together. This mirrors real-world migration statistics: mega-city corridors like London-Paris or New York-Los Angeles typically carry far more absolute migrant volume than routes between small towns, even though the migration rate (flow relative to population) may be similar or even lower for the mega-city pair.
Not necessarily. The simplified gravity model shown here computes a single symmetric exchange volume between each city pair, but real migration is typically asymmetric — more people might move from city A to city B than the reverse, driven by relative economic opportunity, wages, or amenities. This simulation adds a directional bias so that particles are more likely to travel toward the larger of the two cities in a pair, approximating the real-world tendency of migrants to move toward bigger labor markets, while still keeping the underlying M_ij magnitude symmetric for simplicity.
Current research extends the basic gravity model with machine-learning techniques that learn flexible, nonlinear "resistance" functions instead of a fixed power law, incorporate network-based diaspora and social-tie effects, and use radiation models (an alternative to gravity models that considers intervening opportunities rather than raw distance). Researchers also apply gravity-style models to non-human flows such as international trade, airline passenger traffic, and even animal migration, since the same size-times-distance-decay logic recurs whenever movement volume depends on the attractiveness of a destination balanced against the friction of separation.