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.

iterative

Logical scalar if iterative approximation should be used.

num.iter

Number of iterations to be used. defaults to 10 (see Details).

## Value

a matrix containing approximation of relative 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.

## References

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.

David Schoch

## 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    0
approx_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 0.0000000
approx_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