Source-linked AI summary
Systematic comparison of trip distribution laws and models
Maxime Lenormand, Aleix Bassolas, José J. Ramasco
TL;DR
The paper addresses how gravity and intervening-opportunity laws should be compared for estimating commuting flows. It systematically tests these laws with different network constraints and empirical datasets, finding that gravity generally performs better while remaining limited for long-distance flows.
Problem
Existing comparisons of gravity and intervening-opportunity laws use differing inputs, parameters, and constraints, motivating a systematic comparison for commuting-flow estimation.
Method
The study compares multiple gravity and intervening-opportunity laws across commuting data, separates probability laws from constrained trip-generation models, and examines calibration from geographic scale.
Results
Gravity globally outperforms intervening-opportunity approaches for estimating commuting flows, preserving network structure, and fitting commuting distances, with exponential decay performing best overall.
Takeaways & Limitations
Distance appears more important than intervening opportunities in work-location choices, while gravity’s flexible distance-decay function supports its performance in this setting.
Takeaways & Limitations
Gravity fails to estimate commuting flows at large distances, which represent less than 6% of commuters in France and 5% in the United States.
Abstract
from arXiv · showhide
Trip distribution laws are basic for the travel demand characterization needed in transport and urban planning. Several approaches have been considered in the last years. One of them is the so-called gravity law, in which the number of trips is assumed to be related to the population at origin and destination and to decrease with the distance. The mathematical expression of this law resembles Newton's law of gravity, which explains its name. Another popular approach is inspired by the theory of intervening opportunities which argues that the distance has no effect on the destination choice, playing only the role of a surrogate for the number of intervening opportunities between them. In this paper, we perform a thorough comparison between these two approaches in their ability at estimating commuting flows by testing them against empirical trip data at different scales and coming from different countries. Different versions of the gravity and the intervening opportunities laws, including the recently proposed radiation law, are used to estimate the probability that an individual has to commute from one unit to another, called trip distribution law. Based on these probability distribution laws, the commuting networks are simulated with different trip distribution models. We show that the gravity law performs better than the intervening opportunities laws to estimate the commuting flows, to preserve the structure of the network and to fit the commuting distance distribution although it fails at predicting commuting flows at large distances. Finally, we show that the different approaches can be used in the absence of detailed data for calibration since their only parameter depends only on the scale of the geographic unit.
DATA
The study analyzes commuting flows across eight case studies covering six countries and two cities, using census units ranging from small urban areas to large counties.
- Datasets: The case studies cover England and Wales, France, Italy, Mexico, Spain, the United States, London, and Paris.The country datasets use census or census-based commuting data from different years and statistical agencies.
- Spatial scale: The spatial units range from London Output Areas averaging 1.68 km2 to United States counties averaging 2596.8 km2.This provides multiple geographic scales for comparing trip distribution approaches.
- Spatial representation: Figures 1 and 2 show the centroids of the census units used in the eight case studies.Figure 2 maps unit centroids around London and Paris against the Greater London Authority and Île-de-France boundaries.
- Variables: For each unit pair, the data include commuting trips, great-circle distance, and population.Trips count individuals living in origin unit i and working in destination unit j; distance is computed with the Haversine formula.
- Scope: The analysis considers only inter-unit flows, setting self-flows Tii to zero because intra-unit flows cannot be estimated with radiation laws.Out-commuters and in-commuters are defined from the inter-unit trip totals.
COMPARISON OF TRIP DISTRIBUTION LAWS AND MODELS
The paper separates the probability law governing origin–destination choices from the constrained model that allocates the total number of commuting trips.
- Model purpose: Trip distribution models split the total number of trips N into an estimated origin–destination trip table.The table contains estimated trips from each census area to every other area.
- Trip distribution law: A trip distribution law calculates the probability pij of observing a trip between units i and j.This probability is asymmetric because commuting flows are directional, and self-loops are excluded.
- Normalization: The probability pij is normalized across all possible origin–destination pairs so that their total equals 1.It is not the conditional probability that a trip starting in i finishes in j.
- Probability distinction: The analysis distinguishes the unconditional pair probability pij from conditional destination-choice probabilities used by intervening-opportunity laws.The conditional form depends on origin population, destination population, and intervening opportunities.
Gravity laws
The paper compares gravity and intervening-opportunity formulations for commuting probabilities, including exponential and power distance decay and radiation-based alternatives.
- Gravity laws: Gravity probabilities are proportional to origin and destination populations and inversely related to travel cost.The distance effect is represented through a distance-decay function.
- Gravity laws: The study tests both exponential and power distance-decay functions, with parameter β adjusting the importance of distance.The form of the decay function may vary across datasets.
- Intervening opportunities: Intervening-opportunity probabilities combine origin population with the conditional probability of choosing a destination given opportunities lying between origin and destination.The conditional probability is normalized so trips end within the region of interest.
- Intervening opportunities: In Schneider’s formulation, sij counts opportunities within a circle centered at the origin, while γ represents the constant probability of accepting an opportunity destination.The origin opportunities mi are not included in this version.
- Radiation laws: Radiation models describe destination choice through absorption of opportunities and include an original parameter-free version plus an extended version with parameter α.The extended model controls how intervening job opportunities affect job selection.
Constrained models
After defining trip probabilities, the study generates commuting networks by sampling trips under four levels of production and attraction constraints.
- Network generation: The models generate a commuting network by drawing N trips from the trip distribution law.Different models impose different constraints on trip production and attraction.
- Unconstrained model: The unconstrained model preserves only the total number of trips N.Trips are sampled from a multinomial distribution.
- Single constraints: The production-constrained model preserves the number of trips produced by each origin unit.For each unit i, Oi trips are generated from a multinomial distribution.
- Single constraints: The attraction-constrained model preserves the number of trips attracted by each destination unit.For each unit j, Dj trips are assigned from a multinomial distribution.
- Doubly constrained model: The doubly constrained model preserves both production and attraction totals using balancing factors calibrated by iterative proportional fitting.Unlike the other models, it is deterministic and produces real-valued flows in a fully connected network.
Goodness-of-fit measures
The study evaluates commuting predictions with CPC, CPL, and CPCd, complemented by normalized RMSE and information gain. These measures assess flow agreement, network-link overlap, and commuting-distance similarity.
- Goodness-of-fit measures: CPC measures agreement between simulated and observed commuter networks, ranging from 0 for no agreement to 1 for identical networks.The indicator is based on the Sørensen index and is simplified because the total number of commuters is preserved.
- Goodness-of-fit measures: The analysis also uses normalized root mean square error and information gain to test whether conclusions are robust to the goodness-of-fit measure.
- Goodness-of-fit measures: CPL measures the proportion of links shared by simulated and observed networks, with 0 indicating no common links and 1 topological equivalence.
- Goodness-of-fit measures: CPCd compares observed and simulated commuting-distance distributions by applying CPC to distance-bin counts rather than origin-destination flows.The distance bins contain individuals commuting within successive ranges defined by powers of two kilometres.
- Goodness-of-fit measures: Figure 3 compares common commuter shares across eight case studies for normalized gravity, Schneider, and original or extended radiation laws.Symbols distinguish exponential and power gravity decay, Schneider’s law, and the two radiation variants; error bars span 100 realizations.
RESULTS
The study compares five trip-distribution laws across eight case studies and four constrained models, calibrating parameters to maximize CPC. Stochastic simulations are summarized with average CPC values.
- RESULTS: Five laws are tested: exponential- and power-decay gravity, Schneider’s intervening opportunities law, and original and extended radiation laws.
- RESULTS: The evaluation covers eight case studies and four constrained models, with β, γ, and α calibrated to maximize CPC.
- RESULTS: 0.09% is the reported variation around the average for the stochastic simulations.
Estimation of commuting flows
Across the tested models and case studies, normalized gravity generally estimates commuting flows better than intervening-opportunities laws, especially with exponential distance decay. Model constraints substantially improve flow agreement, while alternative metrics largely support the CPC-based findings.
- Estimation of commuting flows: Gravity laws generally outperform intervening-opportunities laws, with exponential decay improving results over power decay; extended radiation slightly exceeds original radiation and Schneider’s law.In unconstrained models, radiation variants can appear better because their normalization makes origin probabilities proportional to origin population.
- Estimation of commuting flows: Figure 4 compares CPC, CPL, and CPCd across unconstrained, production constrained, attraction constrained, doubly constrained, and uniform models.
- Estimation of commuting flows: The comparison uses normalized gravity laws when referring to the gravity approach.
- Estimation of commuting flows: More constraints increase average CPC from around 45% for unconstrained models to 65% for doubly constrained models.The attraction constrained model performs better than the production constrained model in the reported comparison.
- Estimation of commuting flows: The normalized root mean square error and information gain produce results very similar to CPC, although extended radiation has smaller normalized RMSE than normalized gravity in unconstrained models.
Structure of the commuting network
Gravity with exponential distance decay best preserves commuting-network structure in unconstrained and singly constrained models, while link-count biases explain limited overlap. Distance-distribution fit is generally strong but degrades at longer distances.
- Structure of the commuting network: Exponential gravity outperforms other laws for common links in unconstrained and singly constrained models.Under doubly constrained models, results are very similar except for Schneider’s law and original radiation.
- Structure of the commuting network: The common part of links never exceeds 0.55 because the laws fail to reproduce the observed number of links globally.
- Structure of the commuting network: Radiation and exponential gravity tend to underestimate link counts, while extended radiation and power gravity tend to overestimate them.Schneider’s law produces link counts closer to the observed values than the other laws.
Commuting distance distribution
The exponential gravity law reproduces commuting distances slightly better overall, especially at shorter ranges, but fails for the small minority of long-distance commuters.
- More than 80% of commuting distances are reproduced by the gravity and intervening opportunities laws, except the original radiation law.
- The exponential gravity law is better below 50 km in France and 150 km in the United States.
- Beyond those thresholds, the exponential gravity law and Schneider’s law fail to estimate commuting flows accurately.
- Radiation laws and power-law gravity estimate large-distance flows, although such commuters represent less than 6% in France and 5% in the United States.The authors question whether these long trips are daily commutes or artifacts of census collection.
Robustness against changes in the inputs
Parameter estimates are generally robust to input changes for gravity and extended radiation laws, and can often be inferred from geographic scale; Schneider’s law is notably more sensitive.
- Average CPC, CPL, and CPCd results remain stable under input changes across the four models and eight case studies.
- For exponential gravity, β decreases with spatial scale because average commuting distance increases as smaller-distance trips are excluded.
- For power-law gravity, β increases with scale to fit the tail of the commuting-distance distribution as its steepness increases.
- The relationship between Schneider-law γ and spatial scale is not significant, while radiation α increases with average unit surface.
- Gravity-law CPC errors remain at most 4% for exponential decay and 10% for power decay when inputs change.Extended radiation errors are mostly below 10% and reach at most 22%.
- The estimated parameter value can be inferred from the average census-unit surface for gravity and extended radiation laws, enabling calibration without detailed data.
- Schneider’s law is highly sensitive to the surrogate input: estimating γ from population-based regressions causes CPC errors to increase dramatically for in/out-flow inputs.
DISCUSSION
The study finds that gravity models, especially exponential decay, generally outperform intervening-opportunity laws across commuting-flow, network-structure, and distance-distribution measures. Their calibration can usually rely on geographic scale, but long-distance commuting remains a boundary for exponential gravity.
- DISCUSSION: Gravity models generally outperform intervening-opportunity laws in estimating flows, preserving network structure, and fitting commuting-distance distributions.
- DISCUSSION: Exponential gravity performs best overall because most trips are short-range, although it fails to estimate large-distance flows.
- DISCUSSION: Gravity’s superiority is robust to the goodness-of-fit measure and changes in input data.
- DISCUSSION: Except for Schneider’s law, parameters can be estimated from average unit surface, allowing commuting flows to be estimated without detailed calibration data.
- DISCUSSION: Distance appears more important than intervening opportunities in work-location choices, while further research is needed for migration, tourism, and freight movements.
- DISCUSSION: The study emphasizes separating probabilistic laws from constraint levels when comparing spatial-interaction models for planning applications.