Extended Gravity Centrality

Extended gravity centrality (Ma et al. 2016) is the sum of the gravity scores of a node's neighbors, $$G^+(i) = \sum_{j \in N(i)} G(j), \qquad G(j) = \sum_{l:\, 0 < d_{jl} \le r} \frac{k_s(j)\, k_s(l)}{d_{jl}^2},$$ where $k_s$ is the k-shell index and $d$ the hop distance.

Usage

centrality_extended_gravity(x, gravity_radius = 3, ...)

Arguments

x: Network input accepted by centrality.
gravity_radius: Hop radius $r$: a nonnegative number (default 3, the value of Ma et al. 2016), "auto", or NULL or Inf for the whole component.
...: Further arguments to centrality, such as normalized.

Value

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

Details

The measure uses the simple undirected skeleton, so direction, weights, loops and parallel edges are ignored, and the masses are always k-shell indices. The radius applies around each neighbor $j$, so a contribution can come from $r+1$ hops away from $i$, including paths back to $i$. Radius 0 and isolated nodes give 0. The "auto" radius is half the mean finite positive hop distance, rounded to the nearest integer with a minimum of 1. This rule is a package choice. A negative radius raises an error.

References

Ma, L. L., Ma, C., Zhang, H. F., & Wang, B. H. (2016). Identifying influential spreaders in complex networks based on gravity formula. Physica A, 451, 205-212. doi:10.1016/j.physa.2015.12.162 .

Examples

centrality_extended_gravity(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>        504        600        696        588        492        516        432 
#>   Evaluate     Create      Share 
#>        540        612        540