Three Basic Types of Interaction Models

The top part of the figure presents the general formulation of the spatial interaction model where T_ij is the interaction between location i (origin) and location j (destination), V_i are the attributes of the location of origin i, W_j are the attributes of the location of destination j, and S_ij are the attributes of separation between the location of origin i and the location of destination j. From this formulation, three basic types of interaction models can be elaborated:

• Gravity model. The level of interaction between two locations is measured by multiplying their attributes, which is then pondered by their level of separation. Separation is often squared to reflect the growing friction of distance. 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 (again squared to reflect the friction of distance). On the above figure, the potential interaction of location i (T_i) 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 (B_ij) is at a distance of 4.9 from i (and consequently at a distance of 2.1 from j).