DK-Based Gravity Model

The DK-based gravity model (Li and Huang 2021) sums, over the partners within a hop-distance radius $R$, the product of the two nodes' masses divided by their squared distance. The mass $DK(i) = k(i) + k_s^*(i)$ adds the degree to an improved k-shell index $k_s^*(i) = k_s(i) + p(i)/(\max_k q(k) + 1)$, where $p(i)$ is the peeling stage at which the node leaves its shell and $q(k)$ the number of stages shell $k$ needs. $$DKGM_i = \sum_{j \ne i,\; d(i,j) \le R} \frac{DK(i)\,DK(j)}{d(i,j)^2}$$

Usage

centrality_dkgm(x, dkgm_radius = 2, ...)

Arguments

x: Network input accepted by centrality.
dkgm_radius: Hop-distance radius $R$, a nonnegative number (default 2, a value the source recommends). NULL or Inf includes every reachable partner. "auto" uses half the mean finite hop distance, rounded to the nearest integer and at least one, following the source's equation 4.
...: Further arguments to centrality, such as normalized (divide by the maximum, default FALSE).

Value

A named numeric vector with one score per node, in input node order.

Details

The measure is computed on the simple undirected skeleton of the network, so direction, weights, loops and parallel edges are ignored. The k-shell peeling removes nodes of degree at most $k$, which is the reading that terminates and reproduces the source's Tables 2 to 5, so an isolate falls in the one-shell. Unreachable partners contribute nothing, and isolates and a single-node graph score zero. The stage denominator $\max_k q(k) + 1$ is a global maximum, so adding a disconnected component can change every score. A dkgm_radius below one gives zero at every node.

References

Li, Z. and Huang, X. (2021). Identifying influential spreaders in complex networks by an improved gravity model. Scientific Reports, 11, 22194. doi:10.1038/s41598-021-01218-1 .

Examples

centrality_dkgm(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>     580.80     726.30     875.56     716.58     557.89     588.00     452.23 
#>   Evaluate     Create      Share 
#>     603.84     737.64     617.40