Freeman betweenness centrality of a node

C_B(v) = sum_{s != v != t} sigma_st(v) / sigma_st, where sigma_st counts shortest s-t paths and sigma_st(v) those passing through v. Accumulated by Brandes' (2001) recursion.

Usage

Netbtw(A, node = 0)

Arguments

A

Adjacency matrix; non-zero entries are edges (unweighted).

node

Zero-based index of the node whose centrality is returned.

Value

List with estimate, normalized, cb_ordered, node, n, symmetric.

Details

Brandes accumulates over ORDERED pairs; Freeman's C_B is defined over unordered pairs, so for a symmetric adjacency the ordered sum is halved. The normalised score divides by the maximum (n-1)(n-2)/2 attained by the hub of a star.

References

Freeman (1977), Sociometry 40(1), 35-41, doi:10.2307/3033543 ; Brandes (2001), J. Math. Sociol. 25(2), 163-177, doi:10.1080/0022250X.2001.9990249 .

Examples

A <- matrix(c(4, 1, 0.5, 1, 3, 0.8, 0.5, 0.8, 2), nrow = 3)
res <- Netbtw(A = A)
res
#> $estimate
#> [1] 0
#> 
#> $normalized
#> [1] 0
#> 
#> $cb_ordered
#> [1] 0
#> 
#> $node
#> [1] 0
#> 
#> $n
#> [1] 3
#> 
#> $symmetric
#> [1] 1
#> 
#> $method
#> [1] "Freeman betweenness centrality"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"