The Geography of Transport Systems
Jean-Paul Rodrigue (2017), New York: Routledge, 440 pages.
ISBN 978-1138669574
Spatial Interactions and the Gravity Model
Author: Dr. Jean-Paul Rodrigue
1. Overview
One methodology of particular importance to transport geography relates to how to estimate flows between locations, since these flows, known as spatial interactions, enable to evaluate the demand (existing or potential) for transport services.
A spatial interaction is a realized movement of people, freight or information between an origin and a destination. It is a transport demand / supply relationship expressed over a geographical space.
Spatial interactions cover a wide variety of movements such as journeys to work, migrations, tourism, the usage of public facilities, the transmission of information or capital, the market areas of retailing activities, international trade and freight distribution. These movements can be physical (people or freight) or intangible (information). Economic activities are generating (supply) and attracting (demand) flows. 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 an 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 industrial zone because the first is supplying workers while the second is supplying jobs. The same can be said concerning the complementarity between a store and its customers and between an industry and its suppliers (movements of freight).
  • Intervening opportunity. 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.
  • Transferability. Freight, persons or information being transferred must be supported by transport infrastructures, implying that the origin and the destination must be linked. Costs to overcome distance must not be higher than the benefits of related interaction, even if there is complementarity and no alternative opportunity.
Spatial interaction models seek explain 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, it is possible for example to better allocate transport resources such as highways, buses, airplanes or ships since they would reflect the transport demand more closely.
2. Origin / Destination Matrices
Each spatial interaction, as an analogy for a set of movements, is composed of an origin / destination pair. Each pair can itself 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, or a spatial interaction matrix.
O/D Matrix
O/D Pair Destinations
A B C Total
Origins A       Ti
Total Tj     T
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 equals to the summation of outputs. Otherwise, there are movements that are coming from or going to 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 the age group, income, gender, etc. Under such circumstances they are labeled sub-matrices since they account for only a share of the total flows.
In many cases where spatial interactions information is relied on for planning and allocation purposes, origin / destination matrices are not available or are incomplete. Palliating this lack of data commonly requires surveys. With economic development, the addition of new activities and transport infrastructures, spatial interactions have a tendency 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 efforts, time and costs. In a complex spatial system such as a region, O/D matrices tend to be quite large. For instance, the consideration of 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 is likely to rapidly become obsolete as economic and spatial conditions change. It is therefore important to find a way to estimate as precisely as possible spatial interactions, particularly when empirical data is lacking or is incomplete. A possible solution relies on using a spatial interaction model to complement and even replace empirical observations.
3. Spatial Interaction Models
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 concerned origins and the destinations. The general formulation of the spatial interaction model is as follows:
  • Tij : Interaction between location i (origin) and location j (destination). Its units of measurement are varied and can involve people, tons of freight, traffic volume, etc. It also relates to a time period such as interactions by the 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 gross domestic product.
  • Wj : Attributes of the location of destination j. It uses similar socio-economic variables than the previous attribute.
  • Sij : Attributes of separation between the location of origin i and the location of destination j. Also known as transport friction. 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.
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 than 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:
  • Pi and Pj : Importance of the location of origin and the location of destination.
  • dij : Distance between the location of origin and then 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 comparatively 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:
  • P, d and k refers 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 services by a highway will have a lower beta index than if they were serviced by a 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 for retailing flows, a location having higher income levels will generate more movements.
  • 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 movements.
A significant challenge related to the usage of spatial interaction models, notably the gravity model, is related to their calibration. Calibration consists in finding the value of each parameters of the model (constant and exponents) to insure that the estimated results are similar to the observed flows. If it is not the case, the model is almost useless as it predicts or explains little. It is impossible to know if the process of calibration is accurate without comparing estimated results with empirical evidence.
In the two formulations of the gravity model that have been introduced, the simple formulation offers a 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 in 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 a growth in the mobility.
Often, a value of 1 is given to the parameters, and then they are progressively altered until the estimated results are similar to observed results. 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 process, but a very useful one. Once a spatial interaction model has been validated for a city or a region, it can then 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.