connectivity {PlackettLuce}

Check Connectivity of Rankings

Description

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

Usage

connectivity(x, verbose = TRUE)

Arguments

x

an adjacency matrix as returned by adjacency, a "rankings" object, or an object that can be coerced by as.rankings.

verbose

logical, if TRUE, a message is given if the network is not strongly connected.

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.

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

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)
res$membership
## keep items in cluster 1
na.omit(X[,res$membership == 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)
res$membership

## 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))
res$membership

## 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))
res$membership