Compute the Nystuen and Dacey graph for a spatial interaction model

This function computes the most important flows in a spatial interaction model according to the approach outlined by J. D. Nystuen and M. F. Dacey (Nystuen & Dacey, 1961). In this work, a nodal flow is the largest flow sent from a non-terminal location (based on the definition of terminals recalled in terminals()). The nodal structure is the collection of those flows. They form a directed graph that has interesting properties. In particular, each weakly connected component contains a single terminal location which can be seen as the dominant location of the component. Note that because nodal flows are based on terminals, this function applies only to the non-bipartite setting.

Usage

nd_graph(sim, definition = c("ND", "RW"), ...)

# S3 method for class 'sim'
nd_graph(sim, definition = c("ND", "RW"), ...)

Arguments

sim: a non-bipartite spatial interaction model object
definition: terminal definition, either "ND" (for Nystuen & Dacey, default) or "RW" (for Rihll & Wilson), see details.
...: additional parameters

Value

a data frame describing the Nystuen and Dacey graph a.k.a. the nodal structure of a spatial interaction model

Details

In practice, the function first computes the terminals and non-terminals according to either Nystuen & Dacey (1961) or Rihll and Wilson (1991). Then it extracts the nodal flows. The result of the analysis is returned as a data frame with three columns:

from: the index of the non-terminal origin location
to: the index of the destination location of the nodal flow of from
flow: the value of the nodal flow

An important aspect of the nodal structure is that it does not contain isolated terminals. If a location is a terminal but never receives a nodal flow, it does not appear in the collection of nodal flows. It constitutes a trivial connected component in itself.

References

Nystuen, J.D. and Dacey, M.F. (1961), "A graph theory interpretation of nodal regions", Papers and Proceedings of the Regional Science Association 7: 29-42. doi:10.1007/bf01969070

Rihll, T., and Wilson, A. (1991), "Modelling settlement structures in ancient Greece: new approaches to the polis", In City and Country in the Ancient World, Vol. 3, Edited by J. Rich and A. Wallace-Hadrill, 58-95. London: Routledge.

Examples

distances <- french_cities_distances[1:10, 1:10] / 1000 ## convert to km
production <- rep(1, 10)
attractiveness <- rep(1, 10)
model <- blvim(distances, production, 1.3, 1 / 250, attractiveness,
  bipartite = FALSE
)
destination_names(model) <- french_cities$name[1:10]
nd_graph(model)
#>            from to      flow
#> Marseille     2  7 0.9488844
#> Lyon          3  7 0.7594037
#> Toulouse      4  7 0.9046308
#> Nice          5  7 0.9488844
#> Nantes        6  1 0.7793698
#> Strasbourg    8  1 0.6655188
#> Bordeaux      9  7 0.7106013
#> Lille        10  1 0.9291846
dist_times <- french_cities_times[1:15, 1:15]
tmodel <- blvim(dist_times, rep(1, 15), 1.3, 1 / 5000, rep(1, 15),
  bipartite = FALSE
)
destination_names(tmodel) <- french_cities$name[1:15]
terminals(tmodel, definition = "RW")
#>      Paris  Marseille       Lyon   Toulouse Strasbourg   Bordeaux     Rennes 
#>          1          2          3          4          8          9         11 
nd_graph(tmodel, "RW")
#>               from to      flow
#> Nice             5  2 0.9620972
#> Nantes           6 11 0.5920218
#> Montpellier      7  2 0.8707174
#> Lille           10  1 0.9551241
#> Toulon          12  2 0.9727568
#> Reims           13  1 0.9308482
#> Saint-Étienne   14  3 0.5062502
#> Le Havre        15  1 0.8529081