A.18 – Spatial Interactions and the Gravity Model

Author: Dr. Jean-Paul Rodrigue

A spatial interaction is a realized flow of passengers or freight between an origin and a destination. It is a transport demand / supply relationship expressed over a geographical space.

1. Conditions for Spatial Flows

Estimating flows between locations is a methodology of relevance to transportation. These flows, known as spatial interactions, enable to evaluate the demand (existing or potential) for transport services. They cover forms of mobility such as journeys to work, migrations, tourism, public facilities usage, information or capital transmission, retailing activities market areas, international trade, and freight distribution. Mobility can be physical (passengers or freight) or intangible (information), and each form of mobility is subject to a form of friction.

Economic activities are generating (supply) and attracting (demand) movements. The simple fact that a movement occurs between an origin and a destination underlines that the costs incurred by a spatial interaction are lower than the benefits derived from such an interaction. As such, a commuter is willing to drive one hour because this interaction is linked to income, while international trade concepts, such as comparative advantages, underline the benefits of specialization and the ensuing generation of trade flows between distant locations.

Three interdependent conditions are necessary for a spatial interaction to occur:

  • Complementarity. There must be a supply and a demand between the interacting locations. A residential zone is complementary to an employment zone because the first supplies workers while the second supplies jobs. The same can be said concerning the complementarity between a store and its customers and between an industry and its suppliers (freight movements). An economic system is based on a large array of complementary activities.
  • Intervening opportunity (lack of). Refers to a location that may offer a better alternative as a point of origin or as a point of destination. For instance, in order to have an interaction of a customer to a store, there must not be a closer store that offers a similar array of goods. Otherwise, the customer will likely patronize the closer store, and the initial interaction will not occur.
  • Transferability. Transport infrastructures must support mobility, implying that the origin and destination must be linked. Costs to overcome distance must not be higher than the benefits of the related interaction, even if there is complementarity and no alternative opportunity.

Spatial interaction models seek to explain existing spatial flows. As such, it is possible to measure flows and predict the consequences of changes in the conditions generating them. When such attributes are known, allocating better transport resources such as conveyances, infrastructure, and terminals is possible.

2. Origin / Destination Matrices

Each spatial interaction, as an analogy for a set of movements, is composed of a discrete origin/destination pair. Each pair can be represented as a cell in a matrix where rows are related to the locations (centroids) of origin, while columns are related to locations (centroids) of destination. Such a matrix is commonly known as an origin/destination matrix (O/D matrix), or a spatial interaction matrix.


In the O/D matrix, the sum of a row (Ti) represents the total outputs of a location (flows originating from), while the sum of a column (Tj) represents the total inputs (flows bound to) of a location. The summation of inputs is always equaling to the summation of outputs. Otherwise, some movements are coming from or going outside the considered system. The sum of inputs or outputs gives the total flows taking place within the system (T). It is also possible to have O/D matrices according to age group, income, gender, etc. Under such circumstances, they are labeled sub-matrices since they account for only a share of the total flows. If the sample is small and disaggregated, it is possible to use a simple list of interactions instead of a matrix. Still, an origin/destination matrix can be constructed out of this list.

In many cases where spatial interaction information is relied on for planning and allocation purposes, origin/destination matrices are unavailable or incomplete. Overcoming this lack of data commonly requires surveys. With economic development, the addition of new activities, and transport infrastructures, spatial interactions tend to change very rapidly as flows adapt to a new spatial structure. The problem is that an origin/destination survey is very expensive in terms of effort, time, and costs. In a complex spatial system such as a region, O/D matrices are quite large. For instance, considering 100 origins and 100 destinations would imply 10,000 separate O/D pairs for which information has to be provided.

In addition, the data gathered by spatial interaction surveys will become obsolete as economic and spatial conditions change. Therefore, it is important to find a way to estimate as precisely as possible spatial interactions, particularly when empirical data is lacking or incomplete. Further, the emergence of ‘big data’ enabled the collection of large amounts of personal mobility information that is possible to convert into flows between spatial units. In such a context, the purpose of a spatial interaction model is to complement and even replace empirical observations through reliable estimates of flows between locations.

3. Spatial Interaction Models

Spatial interaction models are usually the first two steps in the standard four step transportation / land use model, as they estimate the spatial generation and distribution of trips. The basic assumption concerning many spatial interaction models is that flows are a function of the attributes of the locations of origin, the attributes of the locations of destination, and the friction of distance between the concerning origins and the destinations. The general formulation of a spatial interaction model is as follows:

\large T_{ij} = f(V_{i}, W_{j},S_{ij})

  • Tij : Interaction between location i (origin) and location j (destination). Its units of measurement are varied and can involve the number of passengers, tons of freight, traffic volume, etc. It also relates to a time period, such as interactions by hour, day, month, or year.
  • Vi : Attributes of the location of origin i. Variables often used to express these attributes are socio-economic in nature, such as population, number of jobs available, industrial output, or any proxy of the level of economic activity, such as gross domestic product.
  • Wj : Attributes of the location of destination j. It uses similar socio-economic variables to the previous attribute to underline the reciprocity of the locations.
  • Sij : Attributes of separation between the location of origin i and the location of destination j. Also known as transport friction, friction of distance, or impedance. Variables often used to express these attributes are distance, transport costs, or travel time.

The attributes of V and W tend to be paired to express complementarity in the best possible way. For instance, measuring commuting flows (work-related movements) between different locations would likely consider a variable such as working-age population as V and total employment as W. From this general formulation, three basic types of interaction models can be constructed:

  • Gravity model. Measures interactions between all the possible location pairs.
  • Potential model. Measures interactions between one location and every other location.
  • Retail model. Measure the boundary of the market areas between two locations competing over the same market based on the intensity of their respective interactions.

4. The Gravity Model

The gravity model is the most common formulation of the spatial interaction method. It is named as such because it uses a similar formulation to Newton’s law of gravity. Gravity-like representations have been applied in a wide variety of contexts, such as migration, commodity flows, traffic flows, commuting, and evaluating boundaries between market areas. Accordingly, the attraction between two objects is proportional to their mass and inversely proportional to their respective distance. Consequently, the general formulation of spatial interactions can be adapted to reflect this basic assumption to form the elementary formulation of the gravity model:

\large T_{ij} = k \frac{P_{i} P_{j}} {d_{ij}}

  • Pi and Pj : Importance of the location of origin and the location of destination.
  • dij : Distance between the location of origin and the location of destination.
  • k is a proportionality constant related to the rate of the event. For instance, if the same system of spatial interactions is considered, the value of k will be higher if interactions were considered for a year compared to the value of k for one week.

Thus, spatial interactions between locations i and j are proportional to their respective importance divided by their distance. The gravity model can be extended to include several calibration parameters:

\large T_{ij} = k \frac{P_{i}^{\lambda} P_{j}^{\alpha}} {d_{ij}^{\beta}}

  • P, d, and k refer to the variables previously discussed.
  • β (beta): A parameter of transport friction related to the efficiency of the transport system between two locations. This friction is rarely linear as the further the movement, the greater the friction of distance. For instance, two locations serviced by a highway will have a lower beta index than if they were serviced by a regular road.
  • λ (lambda): Potential to generate movements (emissivity). For movements of people, lambda is often related to an overall level of welfare. For instance, it is logical to infer that a location with higher income levels will generate more movements (customers) for retailing flows.
  • α (alpha): Potential to attract movements (attractiveness). Related to the nature of economic activities at the destination. For instance, a center having important commercial activities will attract more flows.

A significant challenge related to the usage of spatial interaction models, notably the gravity model, is related to their calibration. Calibration consists of finding the value of each model parameter (constants and exponents) to ensure that the estimated results are similar to the observed flows, that those results can be replicated, and that changing the parameters would generate valid results. If not the case, the model is of limited use as it predicts or explains little. It is impossible to know if the calibration process is accurate without comparing estimated results with empirical evidence. Consistent calibration makes the model more rigorous and adaptable to other contexts.

In the two formulations of the gravity model that have been introduced, the simple formulation offers good flexibility for calibration since four parameters can be modified. Altering the value of beta, alpha, and lambda will influence the estimated spatial interactions. Furthermore, the value of the parameters can change over time due to factors such as technological innovations, new transport infrastructure, and economic development. For instance, improvements in transport efficiency generally have the consequence of reducing the value of the beta exponent (friction of distance). Economic development is likely to influence the values of alpha and lambda, reflecting growth in mobility.

Calibration can also be considered for different O/D matrices according to age, income, gender, type of merchandise, and modal choice. A part of the scientific research in transport and regional planning aims at finding accurate parameters for spatial interaction models. This is generally a costly and time-consuming but very useful process. Once a spatial interaction model has been validated for a city or a region, it can be used for simulation and prediction purposes, such as how many additional flows would be generated if the population increased or if better transport infrastructures (lower friction of distance) were provided.

Outside the gravity model, other models can be used to measure spatial interactions. Destination choice models are considered an extension of the gravity model that is gaining popularity since they provide a more extensive range of factors explaining the assignment of spatial interactions. While the gravity model assumes that flows are generated as a function of attributes of the origin and destination pondered by impedance functions, the destination choice model allows for additional behavioral attributes to mobility, including income, walkability, the availability of parking, and psychological barriers. The main goal is to explain flows that the standard gravity model does not capture well.

Related Topics


  • Colwell, P. F. (1982) “Central place theory and the simple economic foundations of the gravity model”, Journal of Regional Science, Vol. 22, No. 4, pp. 541-546.
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  • Fotheringham, A.S. (1983) “Some Theoretical Aspects of Destination Choice and their Relevance to Production-Constrained Gravity Models”, Environment and Planning, Vol. 15A, pp. 464-488.
  • Fotheringham, A.S. and M.E. O’Kelly (1989) Spatial Interaction Models: Formulations and Applications. London: Kluwer Academic.
  • Fotheringham, A.S., P. J. Densham, and A. Curtis (1995) “The zone definition problem in location-allocation modeling”. Geographical Analysis, 27 (1), pp. 60-77.
  • Huff, D.L. and G.F. Jenks (1968) “A Graphic Interpretation of the Friction of Distance in Gravity Models”, Annals of the Association of American Geographers, Vol. 58, No. 4, pp. 814–824.
  • Ullman, E.L. (1956) “The Role of Transportation and the Bases for Interaction”, in W.L. Thomas jr. et al. (eds) Man’s Role in Changing the Face of the Earth, Chicago: University of Chicago Press.
  • Zipf, G.K. (1946) “The P1P2/D Hypothesis: On the Intercity Movement of Persons”, American Sociological Review, Vol. 11, pp. 677-686.