Compute Diffusion Matrix Based on Gravity Model

Computes a connectivity matrix \(\pi_{ij}\) between locations using a normalized gravity model.

Usage

calc_diffusion_matrix_pi(D, N, omega, gamma)

Arguments

D

A square numeric distance matrix between locations. Units are kilometers.

N

A numeric vector of population sizes. Length must match nrow(D). May be named.

omega

Numeric exponent for population scaling.

gamma

Numeric exponent for distance decay.

Value

A numeric matrix of diffusion probabilities \(\pi_{ij}\), with diagonals set to NA.

Details

The normalized gravity model has the following form:

$$\pi_{ij} = \frac{N_j^\omega \, d_{ij}^{-\gamma}}{\sum_{j \ne i} N_j^\omega \, d_{ij}^{-\gamma}}, \quad \forall \; i,\, j$$

where \(d_{ij}\) is the distance between locations \(i\) and \(j\), and \(N_j\) is the population or weight of location \(j\). The diagonal of the output matrix is set to NA, as \(\pi_{ij}\) is intended to be used conditionally alongside the departure probability \(\tau_i\).

Examples

if (FALSE) { # \dontrun{
set.seed(123)
D <- matrix(runif(100, 1, 10), nrow = 10)
D <- (D + t(D)) / 2; diag(D) <- 0
N <- runif(10, 100, 1000)
names(N) <- LETTERS[1:10]

pi_mat <- calc_diffusion_matrix_pi(D, N, omega = 0.5, gamma = 2)
} # }