Spatial Interaction Models

There is a large body of literature concerning gravity and spatial interaction models. They are largely concerned with description and sometimes prediction of interaction (flows) between defined regions. They work on the idea of describing interaction between regions as

$$T_{ij} \sim \frac{P_i P_j}{d_{ij}}$$ (1)

where T_ij is the interaction (trips)between regions i and j, $P_{i \, or \, j}$ is a property of region i or j (analogous to mass or gravity), and d_ij is the ``distance'' (spatial or cost-wise) between regions i and j.

These equations are descriptive, similar to general linear models in regression statistics. They are a way to fit observed data to a concise mathematical model with potential predictive capabilities. They are a standard tool for geographical study; several works give excellent descriptions of their formulations and histories (Golledge and Stimson, 1997; Haynes and Fotheringham, 1984; Lowe and Moryadis, 1975; Wilson and Bennett, 1985). Typical use includes descriptions or analysis of travel linkages between regions (Ivy, 1995) or labor migrations (Fik et al., 1992). They have also been used for parameterization of traffic simulations (Cascetta and Cantarella, 1991) and definition of functional regions based on possible interaction (Noronha and Goodchild, 1992).

 One of the major criticisms of gravity models has been what many consider to be a too literal translation of a Newtonian physics model to social science (Haynes and Fotheringham, 1984, page 17). Wilson and Bennett (1985) alleviated part of this doubt by deriving some of the parameters independently through entropy maximization. However, whatever the analytic justification for the parameters, it can still be inappropriate for a spatial representation of a system. They are an inherently static representation of spatial patterns, though many of the processes that it is used to model are quite dynamic (Fik, 1997, page 399). When one is fitting the model to data, one may not know whether the data are long-term averages, a snapshot in time, or a transition between states. This limitation is not always acknowledged by the people using it.

Dendrinos and Sonis (1990) gave a rigid mathematical treatment to general spatial interaction models, and showed that in equations describing even the simplest cases (one population, or stock interacting in two regions) there are many cases where no equilibrium exists. The implications are that many kinds of spatial interaction are capable of chaotic, complex, or unpredictable behavior, even when described in terms of assumed homogeneity that the gravity model implies. This should serve as an important caveat for any attempts to model dynamic spatial processes as static or equilibrium phenomena.

Gravity models and others similar ones have shown themselves to be valuable for fitting data and parameterizing conceptual relationships, but are useful only to the extent that a sufficiently large body of macroscopic system data is available in a form that the modeler can confidently use for extrapolation.