semigroup: Constructing the Semigroup of Relations of Multiple Networks

Description

Function to create the complete semigroup of multiple relations, where the multiplication table can be specified with either a numerical or a symbolic form.

Usage

semigroup(x, type = c("numerical", "symbolic"), cmps, smpl, valued)

Value

An object of ‘Semigroup’ class. The items included are:

gens: array with generator relations
cmps: array with the unique compound relations
ord: dimension of the semigroup
st: vector of the unique string relations
S: multiplication matrix with semigroup of relations (see below)

If the specified type is numerical, then a matrix of semigroup values is given, otherwise the values is returned as a data frame with the strings of the semigroup.

Arguments

x

an array; usually with three dimensions of stacked matrices where the multiple relations are placed

type

semigroup multiplication table to be returned

numerical for a numerical format (default)
symbolic for a symbolic format

cmps

(optional and logical) whether the composite matrices should be also given in the output

smpl

(optional and logical) whether to simplify or not the strings of relations

valued

(logical) whether the semigroup should be with a valued format

Author

Antonio Rivero Ostoic

Warning

For medium size or bigger sets (having e.g. more the 4 relation types), the semigroup construction could take a long time.

Details

A multiple relation can be defined by square matrices of 0s and 1s indicating the presence and absence of ties among a set of actors. If there is more than one relation type, the matrices must preserve the label ordering of its elements and stacked into an object array in order to be effectively applied to this function.

The semigroup, which is an algebraic structure having a set with an associative operation on it, is calculated considering binary matrices only. This means that if the provided matrices are valued, the function will dichotomise the input data automatically. Values higher or equal to a unit are converted to one; otherwise they are set to zero. If not happy, use function dichot to specify a cutoff value for the dichotomization.

Semigroup structures for valued relations apply the max min operation in the composition of generators and strings.

References

Boorman, S.A. and H.C. White, “Social Structure from Multiple Networks. II. Role Structures.” American Journal of Sociology, 81 (6), 1384-1446. 1976.

Boyd, J.P. Social Semigroups. A unified theory of scaling and blockmodelling as applied to social networks. George Mason University Press. 1991.

Pattison, P.E. Algebraic Models for Social Networks. Cambridge University Press. 1993.

Examples

Run this code

# create the data: two binary relations among three elements
arr <- round( replace( array(runif(18), c(3,3,2)), array(runif(18),
       c(3,3,2))>.5, 1 ) )

# optional: put labels
dimnames(arr)[[3]] <- list("n", "m")

# look at the semigroup with numerical format
semigroup(arr)

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