Jean-Paul Rodrigue (2017), New York:
Routledge, 440 pages.
Spatial Interactions and the Gravity Model
Author: Dr. Jean-Paul Rodrigue
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
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.
conditions are necessary for a spatial interaction to occur:
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.
- 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.
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
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:
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:
- 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
- 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
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:
- 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.
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
- 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.
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.
- P, d and k refers to the variables previously
- 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
- 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.