The positive beta-measure of node i is the sum, over its successors j, of one divided by the in-degree of j. Each node with predecessors shares one unit of domination power equally among those predecessors. This is van den Brink and Gilles' BG-measure (1992, definition 2.1), subsequently called the beta-measure (2000, definition 2.1). The negative variant applies the positive measure to the reversed graph (Boldi and Vigna 2014). It sums reciprocal source out-degrees over incoming neighbors.
Arguments
- x
Network input accepted by
centrality.- beta_direction
Either
"positive"(default, credits sources) or"negative"(credits destinations).- ...
Additional arguments to
centrality.normalized = TRUEdivides scores by their maximum; all-zero scores remain zero. This differs from normalizing to unit total mass.
Details
Uses the simple unweighted graph, retaining direction. Loops and duplicate arcs are removed; weights, mode, inversion and cutoff are ignored. Undirected edges represent reciprocal arcs, so both variants coincide with the sum of reciprocal neighbor degrees. This does not implement the separately defined weighted extension of the original paper.
Nodes without successors have positive score zero; nodes without predecessors have negative score zero. Isolates score zero, and empty graphs return no scores. There is no division by a zero degree: every contributing successor has at least one predecessor. Raw positive scores sum to the number of nodes with nonzero in-degree; raw negative scores sum to the number with nonzero out-degree. In disconnected graphs this accounting applies independently to each component.
Dense matrix preparation and evaluation take O(n squared) time and memory. The score is an expected number of predecessor selections, not a probability distribution or a stationary random-walk centrality.
References
van den Brink, R. and Gilles, R. P. (1992). Measuring domination in directed graphs. Tilburg Research Memorandum FEW 565, definition 2.1 and example 2.2, pp. 3-4. van den Brink, R. and Gilles, R. P. (2000). Measuring domination in directed networks. Social Networks, 22, 141-157, definition 2.1. 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(igraph::make_graph("Zachary"))
#> 1 2 3 4 5 6 7 8
#> 5.1944444 2.3625000 2.1569444 1.2236111 0.6458333 1.1458333 1.1458333 0.4402778
#> 9 10 11 12 13 14 15 16
#> 0.5546569 0.1588235 0.6458333 0.0625000 0.2291667 0.4991013 0.1421569 0.1421569
#> 17 18 19 20 21 22 23 24
#> 0.5000000 0.1736111 0.1421569 0.2324346 0.1421569 0.1736111 0.1421569 0.9754902
#> 25 26 27 28 29 30 31 32
#> 0.7500000 0.7000000 0.3088235 0.6921569 0.3254902 0.8421569 0.4532680 1.2046569
#> 33 34
#> 3.7254902 5.7666667
centrality_beta_measure(igraph::make_star(5, mode = "out"),
beta_direction = "negative")
#> 1 2 3 4 5
#> 0.00 0.25 0.25 0.25 0.25