Gravity Centrality

Gravity centrality (Ma et al. 2016) treats node masses as attracting each other with a force that falls with the squared hop distance, summed over the nodes within a radius $r$: $$G(i) = \sum_{j:\, 0 < d_{ij} \le r} \frac{m_i m_j}{d_{ij}^2}.$$ The default uses the k-shell index as mass and $r = 3$ (Ma et al. 2016). Degree mass without truncation is the gravity model of Li et al. (2019, eq. 1), and degree mass with gravity_radius = "auto" is their local gravity model (eq. 2).

Usage

centrality_gravity(
  x,
  mode = "all",
  gravity_mass = "kshell",
  gravity_radius = 3,
  ...
)

Arguments

x: Network input accepted by centrality.
mode: Direction for directed networks: "all" (default), "out" or "in".
gravity_mass: Node mass: "kshell" (default), "degree" or "legacy".
gravity_radius: Largest hop distance included: a number (default 3), "auto", or NULL for the whole network.
...: Further arguments to centrality, such as normalized.

Value

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

Details

Distances are hop counts, so edge weights are ignored. mode sets the direction of both the distances and the degree or k-shell masses, and mode = "all" treats edges as undirected. The "auto" radius is half the mean finite positive distance, rounded to the nearest integer with a minimum of 1 (Li et al. 2019, eq. 5). A radius below 1 gives a score of 0 for every node. gravity_mass = "legacy" with gravity_radius = NULL computes $\sum_j k_j s_j / d_{ij}^2$, with $k_j$ the degree, $s_j$ the k-shell index and no mass on the focal node. This form differs from the formula of Li et al. (2019).

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.

Li, Z., Ren, T., Ma, X., Liu, S., Zhang, Y., & Zhou, T. (2019). Identifying influential spreaders by gravity model. Scientific Reports, 9, 8387.

Examples

centrality_gravity(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>     148.75     163.75     182.50     163.75     145.00     148.75     105.00 
#>   Evaluate     Create      Share 
#>     148.75     167.50     148.75