Compute the potential of a spatial interaction model

The potential is a measure of stability introduced by Osawa, Akamatsu, and Kogure. It is only defined for spatial interaction models produced by the BLV framework. The function returns NA when applied to spatial interaction models for which a potential cannot be computed. The function applies also to a collection of spatial interaction models as represented by a sim_list.

Usage

sim_potential(sim, ...)

Arguments

sim: a spatial interaction model object (an object of class sim) or a collection of spatial interaction models (an object of class sim_list)
...: additional parameters

Value

the scalar value of the potential, or NA when it cannot be computed. In the case of a sim_list the function returns a numeric vector with one value per model.

Details

The potential of a spatial interaction model is given by

$$\frac{1}{\alpha}\sum_{i=1}^n X_i \log\left(\sum_{j=1}^p Z_j^{\alpha}\, \exp(-\beta c_{ij})\right)-\sum_{j=1}^p\kappa_jZ_j,$$

where:

$c$ is the cost matrix (costs())
$X$ is the production constraint vector (production())
$\alpha$ is the return to scale parameter (return_to_scale())
$\beta$ is the inverse of a cost scale parameter (inverse_cost())
$Z$ is the attractiveness vector (attractiveness())
$\kappa$ is the conversion factor vector (sim_conversion())

References

Osawa, M., Akamatsu, T., & Kogure, Y. (2025). "Most likely retail agglomeration patterns: Potential maximization and stochastic stability of spatial equilibria". https://arxiv.org/abs/2011.06778v2

Examples

distances <- french_cities_distances[1:10, 1:10] / 1000 ## convert to km
production <- log(french_cities$population[1:10])
attractiveness <- log(french_cities$area[1:10])
model <- static_blvim(distances, production, 1.5, 1 / 250, attractiveness)
sim_potential(model) ## must be NA
#> [1] NA
dyn_model <- blvim(distances, production, 1.5, 1 / 250, attractiveness)
sim_potential(dyn_model)
#> [1] 335.9337