The potential is a measure of stability introduced by Osawa, Akamatsu, and
Kogure. It is only defined for spatial interaction models produced by 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.
Value
the scalar value of the potential. In the case of a sim_list
the function returns a numerical 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_conversion(model) ## must NA
#> [1] NA