Performs a complete and exact rank analysis of a given partial ranking. This includes rank probabilities, relative rank probabilities and expected ranks.

`exact_rank_prob(P, only.results = T, verbose = F, force = F)`

- P
A partial ranking as matrix object calculated with neighborhood_inclusion or positional_dominance.

- only.results
Logical. return only results (default) or additionally the ideal tree and lattice if

`FALSE`

.- verbose
Logical. should diagnostics be printed. Defaults to

`FALSE`

.- force
Logical. If

`FALSE`

(default), stops the analysis if the partial ranking has more than 40 elements and less than 0.4 comparable pairs. Only change if you know what you are doing.

- lin.ext
Number of possible rankings that extend

`P`

.- mse
Array giving the equivalence classes of

`P`

.- rank.prob
Matrix containing rank probabilities:

`rank.prob[u,k]`

is the probability that u has rank k.- relative.rank
Matrix containing relative rank probabilities:

`relative.rank[u,v]`

is the probability that u is ranked lower than v.- expected.rank
Expected ranks of nodes in any centrality ranking.

- rank.spread
Standard deviation of the ranking probabilities.

- topo.order
Random ranking used to build the lattice of ideals (if

`only.results = FALSE`

).- tree
Adjacency list (incoming) of the tree of ideals (if

`only.results = FALSE`

).- lattice
Adjacency list (incoming) of the lattice of ideals (if

`only.results = FALSE`

).- ideals
List of order ideals (if

`only.results = FALSE`

).

In all cases, higher numerical ranks imply a higher position in the ranking. That is, the lowest ranked node has rank 1.

The function derives rank probabilities from a given partial ranking
(for instance returned by neighborhood_inclusion or positional_dominance). This includes the
calculation of expected ranks, (relative) rank probabilities and the number of possible rankings.
Note that the set of rankings grows exponentially in the number of elements and the exact
calculation becomes infeasible quite quickly and approximations need to be used.
See `vignette("benchmarks")`

for guidelines and approx_rank_relative,
approx_rank_expected, and mcmc_rank_prob for approximative methods.

De Loof, K. 2009. Efficient computation of rank probabilities in posets.
*Phd thesis*, Ghent University.

De Loof, K., De Meyer, H. and De Baets, B., 2006. Exploiting the
lattice of ideals representation of a poset. *Fundamenta Informaticae*, 71(2,3):309-321.

```
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
res <- exact_rank_prob(P)
# a warning is displayed if only one ranking is possible
tg <- threshold_graph(20, 0.2)
P <- neighborhood_inclusion(tg)
res <- exact_rank_prob(P)
#> Warning: P is already a ranking.
#> Expected Ranks correspond to the only possible ranking.
```