The general formulation of the spatial interaction model is stated as Tij , which is the interaction between location i (origin) and location j (destination).Vi are the attributes of the location of origin i, Wj are the attributes of the location of destination j, and Sij are the attributes of separation between the location of origin i and the location of destination j. From this general formulation, three basic types of interaction models can be derived:
- Gravity model. The level of interaction between two locations is a function of their attributes pondered by their level of separation. Separation is often squared to reflect the non-linear friction of distance, but any exponent can be used. On the above figure, two locations (i and j) have a respective “weight” (importance) of 35 and 20 and are at a distance (degree of separation) of 8. The resulting interaction is 10.9, which is reciprocal.
- Potential model. The level of interaction between one location and all the others is measured by the summation of the attributes of each other location pondered by their level of separation, which is squared to reflect the friction of distance. On the above figure, the potential interaction of location i (Ti) is measured by adding the ratio “weight” / squared distance for each other locations (j, k and l). The potential interaction is 3.8, which is not reciprocal.
- Retail model. This model deals with boundaries, instead of interactions. It assumes that the market boundary between two locations is a function of their separation pondered by the ratio of their respective weights. If two locations have the same importance, their market boundary would be halfway between. On the above figure, the market boundary between locations i and j (Bij), which are separated by a distance of 7, is at a distance of 4.9 from i, and therefore at a distance of 2.1 from j.