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 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.

## 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.

## See also

approx_rank_expected, exact_rank_prob, mcmc_rank_prob

## 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,] 0.0000 0.3333 1.0000 1.00 1.0000
#> [2,] 0.6667 0.0000 0.3333 1.00 0.1429
#> [3,] 0.0000 0.6667 0.0000 0.25 1.0000
#> [4,] 0.0000 0.0000 0.7500 0.00 0.5000
#> [5,] 0.0000 0.8571 0.0000 0.50 0.0000

approx_rank_relative(P,iterative = TRUE)

#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.0000 0.6858 1.0000 1.0000 1.0000
#> [2,] 0.3142 0.0000 0.6026 1.0000 0.8104
#> [3,] 0.0000 0.3974 0.0000 0.7130 1.0000
#> [4,] 0.0000 0.0000 0.2870 0.0000 0.5314
#> [5,] 0.0000 0.1896 0.0000 0.4686 0.0000