Approximate relative rank probabilities $$P(rk(u)<rk(v))$$. In a network context, $$P(rk(u)<rk(v))$$ is the probability that u is less central than v, given the partial ranking P.

approx_rank_relative(P, iterative = TRUE, num.iter = 10)

## Arguments

P A partial ranking as matrix object calculated with neighborhood_inclusion or positional_dominance. Logical scalar if iterative approximation should be used. Number of iterations to be used. defaults to 10 (see Details).

## Value

a matrix containing approximation of mutual rank probabilities. relative.rank[i,j] is the probability that i is ranked lower than j

## Details

The iterative approach generally gives better approximations than the non iterative, if only slightly. The default number of iterations is based on the observation, that the approximation does not improve significantly beyond this value. This observation, however, is based on very small networks such that increasing it for large network may yield better results. See vignette("benchmarks",package="netrankr") for more details.

De Loof, K. and De Baets, B and De Meyer, H., 2008. Properties of mutual rank probabilities in partially ordered sets. In Multicriteria Ordering and Ranking: Partial Orders, Ambiguities and Applied Issues, 145-165.

## Examples

P <- matrix(c(0,0,1,1,1,0,0,0,1,0,0,0,0,0,1,rep(0,10)),5,5,byrow=TRUE)
P#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]    0    0    1    1    1
#> [2,]    0    0    0    1    0
#> [3,]    0    0    0    0    1
#> [4,]    0    0    0    0    0
#> [5,]    0    0    0    0    0approx_rank_relative(P,iterative = FALSE)#>           [,1]      [,2]      [,3] [,4]      [,5]
#> [1,] 0.0000000 0.3333333 1.0000000 1.00 1.0000000
#> [2,] 0.6666667 0.0000000 0.3333333 1.00 0.1428571
#> [3,] 0.0000000 0.6666667 0.0000000 0.25 1.0000000
#> [4,] 0.0000000 0.0000000 0.7500000 0.00 0.5000000
#> [5,] 0.0000000 0.8571429 0.0000000 0.50 0.0000000approx_rank_relative(P,iterative = TRUE)#>           [,1]      [,2]      [,3]      [,4]      [,5]
#> [1,] 0.0000000 0.6858378 1.0000000 1.0000000 1.0000000
#> [2,] 0.3141622 0.0000000 0.6026415 1.0000000 0.8103500
#> [3,] 0.0000000 0.3973585 0.0000000 0.7130124 1.0000000
#> [4,] 0.0000000 0.0000000 0.2869876 0.0000000 0.5313942
#> [5,] 0.0000000 0.1896500 0.0000000 0.4686058 0.0000000