Check the connectivity of the network underlying a set of rankings.

## Value

A list with elements

- membership
a labelled vector of indices specifying membership of clusters in the network of items

- csize
the sizes of clusters in the network of items

- no
the number of clusters in the network of items

## Details

Ranked items are connected in a directed graph according to the implied
wins and loses between pairs of items. The wins and losses can be
summarised as an adjacency matrix using `adjacency`

. From this
adjacency matrix, the graph is inferred and it is checked for
connectivity. A message is given if the network is not strongly connected,
i.e. with at least one win and one loss between all partitions of the network
into two groups. Features of clusters in the network are returned - if
the network is strongly connected, all items belong to the same cluster.

## Examples

```
## weakly connected network:
## one win between two clusters
X <- matrix(c(1, 2, 0, 0,
2, 1, 3, 0,
0, 0, 1, 2,
0, 0, 2, 1), ncol = 4, byrow = TRUE)
X <- as.rankings(X)
res <- connectivity(X)
#> Network of items is not strongly connected
res$membership
#> 1 2 3 4
#> 1 1 2 2
## keep items in cluster 1
na.omit(X[,res$membership == 1])
#> [1] "1 > 2" "2 > 1"
## two weakly connected items:
## item 1 always loses; item 4 only wins against item 1
X <- matrix(c(4, 1, 2, 3,
0, 2, 1, 3), nr = 2, byrow = TRUE)
X <- as.rankings(X)
res <- connectivity(X)
#> Network of items is not strongly connected
res$membership
#> 1 2 3 4
#> 3 1 1 2
## item 1 always wins; item 4 always loses
X <- matrix(c(1, 2, 3, 4,
1, 3, 2, 4), nr = 2, byrow = TRUE)
res <- connectivity(as.rankings(X))
#> Network of items is not strongly connected
res$membership
#> 1 2 3 4
#> 1 2 2 3
## all in separate clusters: always 1 > 2 > 3 > 4
## also miscoded rankings and redundant ranking
X <- matrix(c(1, 2, 3, 4,
1, 0, 2, 3,
1, 1, 2, 0,
1, 0, 3, 4,
2, 2, 0, 4,
0, 0, 3, 0,
2, 4, 0, 0), ncol = 4, byrow = TRUE)
res <- connectivity(as.rankings(X))
#> Recoded rankings that are not in dense form
#> Rankings with only 1 item set to `NA`
#> Network of items is not strongly connected
res$membership
#> 1 2 3 4
#> 1 2 3 4
```