The Spatial Interaction Model (SIM) is a static model aiming to describe and predict the analysis of spatial movements, i.e. the processes or spatial flows emerging as result of given spatial configurations. SIM is a model of spatial flows, i.e. flows of people, commodities, capital, information, etc., from some origin i to some destination j.
The general form of a (double-constrained) SIM reads as
represent the total number of flows (physical or virtual) between the origin i
and the destination j
are the stock variables (e.g. population size, workplaces, etc.) in the places of origin and destination;
are the generalized interaction costs; the term β
is the impedance function, measuring separation effects between i
is the cost-sensitivity parameter.
It should be noted that different types of impedance functions can be used, according to the type of spatial structure under analysis (e.g. the negative exponential for homogeneous centres/nodes in the spatial network; the negative power in the presence of large agglomerations/metropolitan areas, etc.: for a review
are balancing factors, equal to:
both derived from the respective additivity conditions:
can be derived as a probabilistic approach based on statistical equilibrium concepts
. Wilson, in fact, demonstrated that SIM (1)
can be derived from a mathematical optimization problem, by maximizing an entropy function, and can thus be seen as an optimum systems solution. The SIM (1)
can be then perceived as the equilibrium state solution in the network of erratic movements. This approach provided a macro-behavioural context to SIMs, given that entropy can be interpreted in terms of a generalized cost function for spatial interaction behaviour