Beta Measure

The beta measure, or BG-index (van den Brink and Gilles 2000), gives node $i$ the sum, over its successors $j$, of one divided by the in-degree of $j$. Each node with predecessors thus shares one unit of domination power equally among them: $$\beta_i = \sum_{j : i \to j} \frac{1}{d^{in}_j}.$$ The negative variant applies the measure to the reversed network (Boldi and Vigna 2014).

Usage

centrality_beta_measure(x, beta_direction = "positive", ...)

Arguments

x: Network input accepted by centrality.
beta_direction: "positive" (default) credits the sources of arcs. "negative" credits their targets.
...: 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 unweighted network with direction kept, so weights, loops and parallel edges are ignored. An undirected edge is a pair of reciprocal arcs, so on an undirected network both variants equal the sum of the reciprocal degrees of the neighbors. A node without successors has positive score zero, and a node without predecessors has negative score zero. The positive scores sum to the number of nodes with nonzero in-degree, and the negative scores sum to the number of nodes with nonzero out-degree.

References

van den Brink, R. and Gilles, R. P. (2000). Measuring domination in directed networks. Social Networks, 22, 141-157. doi:10.1016/S0378-8733(00)00019-8 .

Boldi, P. and Vigna, S. (2014). Axioms for centrality. Internet Mathematics, 10, 222-262. doi:10.1080/15427951.2013.865686 .

Examples

centrality_beta_measure(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>  0.5833333  1.8333333  0.6666667  1.7500000  0.4166667  0.8333333  0.9166667 
#>   Evaluate     Create      Share 
#>  0.7500000  1.2500000  1.0000000