Hubbell Centrality

Hubbell (1965) centrality solves an input-output system in which the score of a node is one plus the attenuated scores of the nodes it sends ties to: $$c = (I - wW)^{-1}\mathbf{1},$$ where \(W\) is the weighted adjacency matrix and \(w\) is hubbell_weight.

Usage

centrality_hubbell(x, hubbell_weight = 0.5, ...)

Arguments

x

Network input accepted by centrality.

hubbell_weight

Attenuation factor \(w\), a positive number (default 0.5).

...

Further arguments to centrality.

Value

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

Details

The system is solvable when the spectral radius of \(wW\) is below one. Otherwise every score is NA with a cograph_undefined_measure warning. A hubbell_weight of zero or below raises a cograph_bad_parameter error. weighted = FALSE gives every edge weight one. The rows of \(W\) are outgoing ties, so on a directed network the score sums attenuated walks that leave the node. centiserve::hubbell() with weights = NULL sets every weight to 1, so it reproduces these values only when the weights are passed explicitly.

References

Hubbell, C. H. (1965). An input-output approach to clique identification. Sociometry, 28(4), 377-399.

See also

centrality_katz, centrality_power, centrality.

Examples

centrality_hubbell(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>   1.458158   2.319798   1.452777   1.586972   1.145412   1.542454   1.418973 
#>   Evaluate     Create      Share 
#>   1.620997   1.761183   1.908969