multiplex (version 2.1)

semiring: Semiring Structures for Balance Theory

Description

A function to construct semiring structures for the analysis of structural balance theory.

Usage

semiring(x, type = c("balance", "cluster"), symclos = TRUE, transclos = TRUE, labels = NULL, k = 2)

Arguments

x

an object of a `Signed' class

type

balance or cluster semiring?

symclos

(logical) apply symmetric closure?

transclos

(logical) apply transitive closure?

labels

(optional) labels for the semiring output

k

length of the cycle or the semicycle

Value

An object of `Semiring' class. The items included are:

Details

Semiring structures are based on signed networks, and this function provides the capabilities to handle either the balance semiring or the cluster semiring within the structural balance theory.

A semiring combines two different kinds of operations with a single underlying set, and it can be seen as an abstract semigroup with identity under multiplication and a commutative monoid under addition. Semirings are useful to determinate whether a given signed network is balanced or clusterable. The symmetric closure evaluates this by looking at semicycles in the system; otherwise the evaluation is through closed paths.

References

Harary, F, Z. Norman, and D. Cartwright Structural Models: An Introduction to the Theory of Directed Graphs. New York: John Wiley & Sons. 1965.

Doreian, P., V. Batagelj and A. Ferligoj Generalized Blockmodeling. Cambridge University Press. 2004.

See Also

signed, as.signed

Examples

## Create the data: two sets with a pair of binary relations 
## among three elements
arr <- round( replace( array( runif(18), c(3 ,3, 2) ), array( runif(18),
       c(3, 3, 2) ) > .5, 3 ) )

## Make the signed matrix with two types of relations
sg <- signed(arr)

## Establish the semiring structure
semiring(sg)