cograph (version 2.7.2)

centrality_gravity: Gravity centrality

Description

\(G(i) = \sum_j m_i m_j / d_{ij}^{2}\), optionally truncated at gravity_radius. The published members of the family differ only in the mass and the reach:

Usage

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

Value

Named numeric vector, one value per node.

Arguments

x

Network input: matrix, igraph, network, cograph_network, or tna object.

mode

Direction: "all", "out" or "in".

gravity_mass

"kshell" (default), "degree", or "legacy".

gravity_radius

Largest distance to include: a number, "auto" for half the mean distance, or NULL for the whole graph. Default 3.

...

Additional arguments passed to centrality.

Change in 2.4.8

Before 2.4.8 this measure computed \(\sum_j k_j s_j / d_{ij}^2\): the product of degree and k-shell on the partner, no mass at all on the focal node, and no truncation. That is not the formula of Li et al. (2019) that its help page cited, and dropping the focal mass changes the ranking rather than the scale. The default is now Ma et al. (2016). gravity_mass = "legacy" with gravity_radius = NULL reproduces the earlier values exactly.

Details

Gravity centrality (Ma, Ma, Zhang & Wang 2016)

k-shell mass, radius 3 -- the default.

Gravity model (Li, Ren, Ma, Liu, Zhang & Zhou 2019, eq. 1)

gravity_mass = "degree", gravity_radius = NULL.

Local gravity model (same paper, eq. 2)

gravity_mass = "degree", gravity_radius = "auto", which uses their empirical half-mean-distance heuristic (eq. 5). cograph rounds to the nearest integer (ties to even), with minimum 1, using finite positive distances on disconnected graphs. These rounding and disconnected-graph rules are cograph conventions.

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.

See Also

centrality_coreness, centrality_kreach, centrality.

Examples

adj <- matrix(0, 6, 6)
adj[cbind(c(1, 1, 2, 4, 4, 5, 3), c(2, 3, 3, 5, 6, 6, 4))] <- 1
adj <- adj + t(adj)
rownames(adj) <- colnames(adj) <- LETTERS[1:6]
centrality_gravity(adj)
centrality_gravity(adj, gravity_mass = "degree", gravity_radius = NULL)