Package {multiplex}

Algebraic Tools for the Analysis of Multiple Social Networks

Description

Algebraic procedures for analyses of multiple social networks are provided with this package as described in Ostoic (2020) <DOI:10.18637/jss.v092.i11>. 'multiplex' supports the creation and analysis of multiplex, multimode, and multilevel network data in various formats. It combines methods based on partially ordered semigroups, decomposition procedures, semiring structures, and relational bundles for multivariate network analysis. It also provides an algebraic approach to affiliation networks using Galois derivations, with visualization options.

Details

One aim of the multiplex package is to provide an analytical tool specifically designed for social networks with relationships at multiple levels. It includes functions for modelling local role algebras based on the simple and compound relations present in the network. multiplex also supports the construction and analysis of signed networks through semiring structures. In addition, it can identify dyadic relational patterns that may support further analysis using a range of structural theories. The package can also incorporate actor attributes into the analysis of multiple networks in different ways. For example, it can assess the network exposure of actors in a multiplex context or embed attribute information directly into the resulting algebraic structures.

A typical workflow starts with function semigroup, which constructs a semigroup from an array of multiple binary relations. The resulting structure can be described with functions wordT and edgeT. Unique relations and associated equations are returned by strings function together with the set of equations with strings of length k. Ordering among string elements is defined by function partial.order, and the corresponding lattice can be displayed with functions hasse or diagram.

Further semigroup analysis is available through function green.rel for equivalence classes and diagram function again for “egg-box” visualization. Structure reduction of semigroup objects is supported through decomposition based on congruence relations or \pi-relations using functions pi.rels, cngr, and decomp.

Signed networks can be created with function signed and analysed with function semiring for balance or cluster semiring structures.

General network analysis utilities in multiplex include function dichot for dichotomization of matrices and arrays, rm.isol for removing isolates, perm for permutation operation, and function zbind for building three-dimensional arrays.

The package multiplex also provides function rbox for Relation-Box construction and function cph for Cumulated Person Hierarchy in the network. Bundle analysis is supported by functions bundles, transf, bundle.census, and summaryBundles. Network exposure and relational-system analysis are available through functions rel.sys and expos, while galois supports algebraic analysis of two-mode networks. Multivariate network data can be imported from send-receive-ties edge lists with function edgel and from formats such as UCINET dl and Visone gml. Multiple network structures can be visualized further with the related package multigraph.

Author(s)

J. Antonio Rivero Ostoic

Maintainer: Antonio Rivero Ostoic <multiplex@post.com>

References

Pattison, P.E. Algebraic Models for Social Networks. Structural Analysis in the Social Sciences. Cambridge University Press. 1993.
Boyd, J.P. Social Semigroups. A unified theory of scaling and blockmodelling as applied to social networks. George Mason University Press. 1991.
Lorrain, F. and H.C. White, “Structural Equivalence of Individuals in Social Networks.” Journal of Mathematical Sociology, 1, 49-80. 1971.
Boorman, S.A. and H.C. White, “Social Structure from Multiple Networks. II. Role Structures.” American Journal of Sociology, 81 (6), 1384-1446. 1976.
Ostoic, J.A.R. Algebraic Analysis of Social Networks. Wiley Series in Computational and Quantitative Social Sciences. Wiley. 2021.

See Also

multigraph, bmgraph, ccgraph.

Examples

# create a three-dimensional array
arr <- round( replace( array(runif(18), c(3,3,2)), array(runif(18),
       c(3,3,2))>.5, 3 ) )

# dichotomize array with customized cutoff value
dichot(arr, c = 3)

# construct the semigroup structure and look at its Green relations
semigroup(arr) |> 
  green.rel()

# look at string relations
strings(arr)

Coerce into a Semigroup Object

Description

A generic function for coercing a relational system into a “Semigroup” class object.

Usage

as.semigroup(x, gens, lbs, numerical, edgeT)

Arguments

Details

Use this function to convert an array representing a semigroup structure into an R object of class “Semigroup” as some routines in multiplex require.

Value

A list object of class “Semigroup” that is either ‘symbolic’ or ‘numerical’.

Class attribute, and a warning message when no generators are provided.

See Also

semigroup, green.rel

Examples

# create a labeled multiplication table data
s <- matrix(data=c(1, 1, 1, 3, 3, 3, 3, 3, 3), nrow=3, ncol=3, byrow=TRUE)

# set attributes to object
attr(s, "dimnames") <- list(1:3, 1:3)

# make 's' a semigroup object
as.semigroup(s)

Coerce into a Signed Object

Description

A generic function for coercing a matrix into a “Signed” class object.

Usage

as.signed(x, lbs)

Arguments

Details

Use this function to convert an array representing a signed network into an R object of class “Signed”, as required by function semiring.

The array in x itself can be also a “Semiring” class object represented as ‘Rel.Q’ attribute for the semiring relational structure.

Value

The input array in x as a “Signed” class.

See Also

signed, semiring

Examples

# load a network dataset
data("incubA")

# coerce parts of the signed matrix with two types of relations
signed(incubA$IM)$s[1:2,1:2] |> 
  as.signed()

Coerce Object into a Strings Class

Description

A generic function for coercing an R object into a “Strings” class.

Usage

as.strings(x, lbs)

Arguments

Details

Use this function to prepare an object for establishing the partial order of relation strings by converting it to class “Strings”. It returns as well the number of unique relations in the semigroup represented by the dimensions in x.

Value

The input array in x as an object of “Strings” class

See Also

strings, partial.order, zbind

Examples

# create the data: two sets with a pair of binary relations among 
# three elements
arr1 <- round( replace( array(runif(18), c(3,3,2)), array(runif(18),
        c(3,3,2))>.5, 3 ) )

arr2 <- round( replace( array(runif(18), c(3,3,2)), array(runif(18),
        c(3,3,2))>.5, 3 ) )

# bind the data sets
arrs <- zbind(arr1, arr2)

# make the data a strings object
as.strings(arrs)

internal function

Description

internal function

Bundle Census of Multiple Networks

Description

A function to perform the bundle census in multiple networks.

Usage

bundle.census(x, loops = FALSE)

Arguments

Details

A bundle is a particular type of pattern made of relations at different levels that is binding a pair of nodes or actors, and this function calculates the number of occurrences for each bundle class pattern in a given multiple network in x. The different types of relations in the network are typically recorded in a three dimensional array of stacked matrices, and edge-list and pairwise-list of relations need to be transformed first into arrays with for example function transf.

Depending on the direction and occurrence of each possible tie, then it is possible to count with seven dyadic configuration classes in the census given in the output where fFunctions bundles and summaryBundles provide bundle class occurrences in the network with a more detailed information.

Value

A table with the number occurrences in the distinct bundle class patterns occuring in x:

• BUNDLES Total number of bundles in x, excluding the Null pattern.

• NULL Null

• ASYMM Asymmetric

• RECIP Reciprocal

• T.ENTR Tie Entrainment

• T.EXCH Tie Exchange

• MIXED Mixed pattern

• FULL Full

• LOOP Loops (if TRUE)

References

Ostoic, J.A.R. “Dyadic Patterns in Multiple Networks,” Advances in Social Networks Analysis and Mining, International Conference on, 475-481. 2011.

See Also

bundles, summaryBundles, transf

Examples

# create two binary relations among three elements
arr <- round( replace( array(runif(18), c(3,3,2)), array(runif(18),
       c(3,3,2))>.8, 3 ) )

# compute the bundle census
bundle.census(arr)

Bundle Class Patterns

Description

Classify bundle class patterns in a system of multiple relations.

Usage

bundles(x, loops = FALSE, smpl = FALSE, lb2lb = TRUE, collapse = FALSE, sep)

Arguments

Details

A bundle is a particular type of pattern made of relations at different levels that is binding a pair of nodes or actors in a network of relationships. A bundle class is a dyadic configuration resulting from the mixture of the direction and the types of ties between the nodes or actors. There are in total seven dyadic configuration classes, which are null, asymmetric, reciprocal, tie entrainment, tie exchange, mixed, and the full bundle pattern. This function provides detailed information about the bundle class patterns in multiple networks as lists of pair relations among the nodes or actors, except for the “null” pattern.

In case that the nodes are not labeled, then an identification number will be assigned according to the location of the nodes in the array representation and as well when the lb2lb option is set to FALSE. This function assumes that the network is directed, and self ties are also considered in the output. Long string labels are simplified with smpl, whereas the collapse option blurs the levels in the strings.

The input array for this function is always dichotomized, and it is possible to obtain the total number of occurrences in each bundle class pattern with the bundle.census function.

Value

An object of “Rel.Bundles” class with the distinct bundle class patterns.

References

Ostoic, J.A.R. “Dyadic Patterns in Multiple Networks,” Advances in Social Networks Analysis and Mining, International Conference on, 475-481. 2011.

See Also

bundle.census, summaryBundles, transf.

Examples

# create two binary relations among three elements
arr <- round( replace( array(runif(18), c(3,3,2)), array(runif(18),
       c(3,3,2))>.8, 3 ) )

# look at bundle patterns
bundles(arr)

Congruence Relations

Description

Find the congruence relations of a given abstract semigroup or a partially ordered semigroup.

Usage

cngr(S, PO, uniq)

Arguments

Details

Congruencies are equivalence relations that preserve the operation between the correspondent classes in the algebraic structure where the different congruence classes are based on the substitution property of the semigroup object.

In case that the partial order is supplied in PO the computation of the congruence classes is slightly faster than for an abstract semigroup.

Value

A “Congruence” class list object with:

References

Hartmanis, J. and R.E. Stearns Algebraic Structure Theory of Sequential Machines. Prentice-Hall. 1966.

See Also

decomp, fact, pacnet

Examples

# create 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 ) )

# abstract semigroup of 'arr'
S <- arr |> semigroup()

# look at the congruences in S
cngr(S)

Find Components in Multiple Networks

Description

Function to find different components and isolated nodes in the multiple networks.

Usage

comps(x, bonds = c("entire", "strong", "weak"), sort)

Arguments

Details

Components and isolates in network x are derived from the transitive closure of bundle ties. By default, the computation takes the entire system, but the bonds option lets the selection of other types of relational bundles for the resulted components.

The sort argument outputs components length in ascending size, such as dyads, triads, and larger groups when present. Changing the order of the existing components may have consequences in the layout of the graph when plotting it with function multigraph.

Value

A list object of length two with the bonds type in the class attribute:

See Also

bundles, summaryBundles, bundle.census, rel.sys

Examples

# create 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 ) )

# find the components and isolates
comps(arr)

internal function

Description

internal function

internal function

Description

internal function

Cumulated Person Hierarchy

Description

A function to calculate the Cumulated Person Hierarchy in networks of multiple relations.

Usage

cph(W, lbs)

Arguments

Details

Uses the cumulated person hierarchy to determine partial structural equivalence among actors in a multiple network. Two nodes are considered partially structurally equivalent only if they have identical role sets. Output is determined by the properties of the Relation-Box product of function rbox, and in case that the length of the labels differs from the order of the relational system, then labels will be ignored.

Value

An object of “Partial.Order” class with an array representing the cumulated person hierarchy.

References

Breiger, R.L. and P.E. Pattison, “Cumulated social roles: The duality of persons and their algebras,” Social Networks, 8, 215-256. 1986.
Mandel, M.J. “Roles and networks: A local approach.” B.A. Honours thesis, Harvard University. 1978.

See Also

rbox, semigroup, diagram

Examples

# load a dataset
data("incubA")

# construct the Relation Box of the dataset image matrices and
# compute its cumulated person hierarchy
incubA |> getElement("IM") |> 
  rbox() |> cph()

Conceptual Scaling for Many-valued Formal Contexts

Description

A function to interpret a given many-valued formal context through a chosen scaling process.

Usage

cscl(x, cl, scale = c("nominal", "ordinal", "intord", "biord", "dichot"), sep)

Arguments

Details

Conceptual scaling is a special form of formal context in which each attribute of a many-valued context is interpreted through a scaling procedure. More broadly, it converts many-valued data into dichotomous data, producing a derived context according to the selected type of scaling. Conceptual scaling is therefore an interpretive method where both the choice of scale and the transformation of values are form part of the analysis, rather than following from a mathematically determined procedure.

The formal context represented in x must be associated to its conceptual scaling-levels recorded in cl as an ordered vector with all levels, and a type of scaling derives the context into a dichotomous data frame with attribute ordinal values. Once the partial order of the Galois derivation among the objects and attributes of the formal concept has been established, the resulting derived context with dichotomous data can be represented as a concept lattice.

Value

A data frame with a “conceptual.scaling” class representing a derived many-valued formal context with respect to the conceptual scaling-levels in cl. The ordered conceptual levels and the pair separator are given in the class attribute.

References

Ganter, B. and R. Wille Formal Concept Analysis – Mathematical Foundations. Springer. 1996.

See Also

galois, partial.order, diagram.

Decomposition of a Semigroup Structure

Description

A function to perform the decomposition of a semigroup structure.

Usage

decomp(S, pr, type = c("mca", "pi", "at", "cc"), reduc, fac)

Arguments

Details

The decomp function reduces an algebraic structure, such as a semigroup, to determine which class members in the system are congruent. The resulting object consists of congruent elements that belong to the lattice of congruence classes in the structure. If the input data comes from the Pacnet program, these elements take the form of \pi-relations or meet-complements of atoms; otherwise, they are equivalent elements that satisfy the substitution property.

A “Semigroup” class object may contain unavailable values in its multiplication table, typically when it is produced as an image by the fact function. In such cases, the semigroup can be reduced with the force option, which adds equations to the string relations to eliminate NA values in the semigroup data.

Use the reduc function to reduce the partial order table.

Value

A list object of “Decomp” class having:

References

Pattison, Philippa E. Algebraic Models for Social Networks. Cambridge University Press. 1993.
Hartmanis, J. and R.E. Stearns Algebraic Structure Theory of Sequential Machines. Prentice-Hall. 1966.

See Also

fact, cngr, reduc, pi.rels, semigroup, partial.order, green.rel.

internal function

Description

internal function

Plot Diagrams of Ordered or Linked Relations

Description

A function to plot and manipulate Hasse and Concept diagrams of ordered relations, or the Egg-box of a semigroup structure.

Usage

diagram(x, type = c("hasse", "concept", "egg-box"), attrs = NULL, main = NULL, 
        incmp, shape = c("rectangle", "circle", "box", "ellipse"), col, col0, 
        fcol, ecol, lty, lbs, ffamily, fstyle, fsize, cex.main, col.main, bg, 
        mar, sep, ...)

Arguments

For "egg-box" type diagram, the following arguments are ignored:

Details

diagram is a wrapper function for plotting and manipulating “Hasse”, “Concept”, and “Egg-box” diagrams.

The first two diagram types represent systems of ordered relations and are displayed as partial or linear order diagrams. For example, the partial order table produced by the ‘strings’ option in the partial.order function can be plotted as a Hasse diagram. Another example comes from Galois derivations in Formal Concept Analysis, where a Concept diagram shows the ordering of formal concepts defined by intents and extents. Roman numerals are rendered for diagram elements when the partial order table in x is not labelled.

On the other hand, the Egg-box “diagram” represents equivalence classes in an abstract semigroup without a partial order structure, and such representation is in the Console, not on the Graphic Device as the other representations.

Value

Depending on the type diagram:

Note

This function requires Rgraphviz package installed.

See Also

hasse, partial.order, strings, galois, green.rel, diagram.levels, as.strings, ccgraph.

Examples

# load a dataset
data("incubA")

# a partial order of strings from dataset image matrix
po <- incubA |> getElement("IM") |> 
  as.strings() |> 
  partial.order(type="strings")

# plot the order relation as a Hasse diagram
## Not run: if(require(Rgraphviz)) {
plot(diagram(po, type="hasse"))
}
## End(Not run)

Levels in Lattice Diagram

Description

A function to represent a textual interpretation of the levels in a given lattice diagram.

Usage

diagram.levels(x, perm = FALSE)

Arguments

Details

This function provides a textual interpretation of the different levels in the lattice diagram of the partial order structure among actors and ties in the network.

When reducing a multiple network, the partial order structure may define different classes of elements based on their inclusion relations. The illustration produced by the diagram function typically shows these levels of ordered relations, which this routine then interprets.

Value

A named list with components of the “levels” in the concept diagram produced by diagram.

If perm is specified, the function returns a data frame containing the elements of the partial order structure, with column names indicating the element classes. It also returns a vector of levels and a matrix of the permuted structure.

Note

This function requires that the Rgraphviz package is installed. Since function grDevices::pictex() inside this routine is for historical interest only since R 4.4.0, the warning message has been suppressed before its future replacement.

See Also

partial.order, diagram, perm

Examples

# load the data
data("incubA")

# given e.g. a partial order table in the object 'po'
po <- incubA |> getElement("IM") |> 
  as.strings() |> 
  partial.order(type="strings")

# find the levels in the lattice diagram
## Not run: 
diagram.levels(po)

## End(Not run)

Dichotomize Data with a Cutoff Value

Description

Function to dichotomize the input data for the semigroup construction with a cutoff value.

Usage

dichot(x, c = 1, diag)

Arguments

Details

This is a wrapper of the replace base function where in this case is aimed to specify a cutoff value for the dichotomization of the data. Values equal or higher to the cutoff are hence converted to 1, while the others are set to 0. Notice that the cutoff value in c can be any real number, and labels are preserved after the dichotomization..

Value

Binary values of the input data with the same format as x.

See Also

replace, mnplx, semigroup.

Examples

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

# dichotomize it with a cutoff value of two
dichot(arr, c = 2)

internal function

Description

internal function

Edge Table Generator

Description

Produce the Edge Table of multiple relations.

Usage

edgeT(x, semigroupClass)

Arguments

Details

Function edgeT is to produce the Edge Table of multiple relations in a network, which is a complete right multiplication table of the semigroup having its elements for each of its generators,

Value

A list of “EdgeTable” class object.

References

Cannon, J.J. “Computing the ideal structure of finite semigroup,” Numerische Mathematik, 18, 254-266. 1971.
Pattison, P.E. Algebraic Models for Social Networks. Cambridge University Press. 1993.

See Also

wordT, semigroup.

Examples

# create 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 ) )

# obtain its edge table
edgeT(arr)

Read and Transform Edge-Lists

Description

A function to read edge lists as objects or files with a ‘sender-receiver-tie’ format representing multiplex networks and to transform edge-lists into three-dimensional arrays.

Usage

edgel(x, toarray = TRUE, header = TRUE, sep = "\t", valued = FALSE, inc = FALSE, 
      rownames = TRUE, add = NULL, na.rm)

Arguments

Details

The edgel function reads edge lists in ‘sender-receiver-tie’ format as a data frame with at least two columns for sender and receiver, plus one column for each relation type in a multiplex network. For simple networks, two columns for sender and receiver are sufficient. The input in x may also be a file containing tabular data, where the separator and header are specified with sep and header.

The inc option is used for incidence matrices with actors or objects and self-ties in data frames, which are relevant for two-mode data and analyses such as Galois transformations between partite sets. Incidence matrices can be converted into diagonal matrices for further analysis. When toarray is set to FALSE, argument rownames can be used to move labels between row names and the first column.

Additional options make it possible to remove missing values with na.rm, add isolated nodes with add, and choose between dichotomizing or preserving values in networks with valued.

Value

By default the outcome is an array that for multiple networks is a three dimensions of stacked matrices where the relation types are placed. However, if toarray is set to FALSE, then the edge-list with a data frame format is returned. warning messages for missing information, with the tranformation type, and when ‘one type of relation assumed’ in x.

Note

For backwards compatibility, an alias for this fuction is read.srt.

See Also

write.edgel, galois, read.gml, read.dl

Network Exposure for Multiple Networks

Description

Function to measure the network exposure of the nodes according to a chosen relational system representing the multiple network.

Usage

expos(rs, classes = FALSE, allClasses = FALSE, allNodes = TRUE)

Arguments

Details

This function generalises the measure of network exposure for multiple networks based on the characteristics of the selected relational system. Such a system may represent the entire network or a configuration of strong or weak ties among the actors. It is also possible to specify different behaviours of the nodes representing social actors, expressed through the relational system. The network exposure measure is calculated on the basis of the immediate neighbours of the reference actor.

Value

References

Ostoic, J.A.R. “Creating context for social influence processes in multiplex networks.” Network Science, 5(1), 1-29.
Valente, T. W. Social networks and health. Oxford University Press. 2010.
Rogers, E. The Diffusion of Innovations. 5th ed. (1st ed. 1964) The Free Press. 2003.

See Also

rel.sys, neighb, bundles

Examples

# create 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 ) )

# compute the exposure measure first array in 'arr' is for attributes with adopter categories
arr |> 
  rel.sys(att = 1) |> 
  expos(classes = TRUE)

Factorize Partially Ordered Semigroups

Description

A function to factorize partially ordered semigroup structures by induced inclusions.

Usage

fact(S, P, uniq = TRUE, fac, atoms, mca, atmc, patm, k)

Arguments

Details

Factorization is a subdirect-representation-based decomposition technique for a partially ordered semigroup. This procedure decomposes the algebraic structure into simpler components, called factors, each of which is a semigroup factor representing a specific homomorphic image of the algebra.

For partially ordered semigroups, these factors are isotone homomorphic images corresponding to maximally independent features of the original semigroup of relations. The quotient semigroups and the reduced partial order tables together form the factorizing set, which is given by the decomposition with decomp.

Value

A list of “Ind.incl” class object having:

Class attribute, and a warning message when no substitution property found in induced inclusions.

References

Ardu, S. ASNET – Algebraic and Statistical Network Analysis. User Manual. University of Melbourne. 1995. (for the implementation algorithm)

See Also

decomp, cngr, pacnet

Examples

# create a partially ordered semigroup
arr <- round( replace( array(runif(18), c(3,3,2)), array(runif(18),
       c(3,3,2))>.5, 1 ) )

# semigroup of relations
S <- semigroup(arr)

# string relations and partial order
P <- strings(arr) |> 
  partial.order()

# perform the factorisation of PO S
fact(S, P)

internal function

Description

internal function

Principal Order Filters

Description

A function to find principal order filters and order ideals in a partial order structure.

Usage

fltr(x, PO, ideal = FALSE, rclos)

Arguments

Details

This function identifies the principal order filter or principal order ideal of an element in a partial order. The input is typically either a concept, an object, or an attribute, together with the corresponding partial order of concepts derived from Galois derivations. If the reference element is a concept, specify it as a positive integer representing the concept label. Alternatively, an object or attribute may be given by its character name, provided it appears among the dimension labels of the partially labelled order table. When the reference element is an object or attribute, principal order filters with full labelling are not supported, and in that case use the corresponding concept number instead.

Value

Returns a named list containing the elements in the upset or downset of the principal order filter or order ideal associated with the reference element in the partial order.

References

Ganter, B. and R. Wille Formal Concept Analysis – Mathematical Foundations. Springer. 1996.

See Also

galois, partial.order, diagram.

Examples

# create a data frame
dfr <- data.frame(x=1:3, y=5:7)

# partial ordering of concepts
PO <- dfr |> 
  galois() |> 
  partial.order(type="galois")

# order filter for the first element
fltr(1, PO, rclos=TRUE)

Galois Derivations

Description

Function to compute Galois derivations between partially ordered subsets.

Usage

galois(x, labeling = c("full", "reduced"), sep)

Arguments

Details

A Galois derivation, or Galois connection, between the power sets of G and M is defined for subsets A \subseteq G and B \subseteq M by the set of attributes common to all objects in A, or A^\prime, and the set of objects having all attributes in B, or B^\prime.

An inclusion with the "full" labeling in the output implies first objects and then attributes in the Galois connection, whereas with the "reduced" option the output assumes inclusions of attributes first and then the objects.

Notice that for many-valued formal contexts the structure is derived from a concept scaling through function cscl where is possible to define a pair separator between attributes and their values. A warning message is produced when there is a conflict between such separators and the pair separator chosen in sep for extents and intents.

Value

A list of “Galois” class object with Galois derivations of objects and attributes as list labels:

Notice that the object attribute class only is printed for "full" labeling, which is an intrinsic part of the outcome for a "reduced" labeling.

References

Ganter, B. and R. Wille Formal Concept Analysis – Mathematical Foundations. Springer. 1996.

See Also

partial.order, cscl, diagram, fltr.

Examples

# incidence matrix as a data frame
dfr <- data.frame(x=1:3, y=5:7)

# compute galois derivations 
galois(dfr)

Green's Relations of Abstract Semigroups

Description

A function to produce the Green's relations of a semigroup object.

Usage

green.rel(x)

Arguments

Details

Function green.rel produces the egg-box diagram (Green, 1951) of an abstract semigroup S, which is the union of the left compatible R equivalence and the right compatible L equivalence classes that makes the D-class on S. Notice that some systems have the D-class equal to S.

Value

A list object of “Green.Rels” and “Semigroup” class having:

References

Green, J. “On the structure of semigroups,” Annals of Mathematics 54(1), 163–172, 1951.
Ostoic, JAR “Relational systems of transport network and provinces in ancient Rome,” in Mathematics for social sciences and arts – algebraic modeling. Springer Nature. 2023.

See Also

semigroup, diagram, as.semigroup, edgeT, wordT, fact, cngr, decomp

Examples

# create 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 string labels
dimnames(arr)[[3]] <- list("n", "m")

# look at the Green's relations of its semigroup
semigroup(arr) |> 
  green.rel()

Hasse Diagram of Set of Ordered Relations

Description

A function to plot the Hasse Diagram of partially ordered relations.

Usage

hasse(x, concept, attrs = NULL, main = NULL, incmp, shape = c("rectangle", 
    "circle", "box", "ellipse"), col, col0, fcol, ecol, lty, 
    lbs, ffamily, fstyle, fsize, cex.main, col.main, bg, mar, 
    sep, ...)

Arguments

Details

A Hasse diagram is a pictorical device to represent systems of partially ordered relations where the hasse function provides arguments for visual manipulation of the diagram. An example of a partially ordered system is the partial order table that is the outcome of the ‘strings’ option in the partial.order function.

Roman numerals are given for elements when the partial order table is not labelled.

Value

A plot of a Hasse diagram with specified settings for a partial or a linear order of relations.

Note

This function requires Rgraphviz package installed.

See Also

diagram, partial.order, strings, galois, green.rel, diagram.levels, as.strings.

Examples

# load a dataset
data("incubA")

# a partial order of strings from dataset image matrix
po <- incubA |> getElement("IM") |> 
  as.strings() |> 
  partial.order(type="strings")

# plot the order relation as a Hasse diagram
## Not run: if(require(Rgraphviz)) {
plot(hasse(po))
}
## End(Not run)

Person and Relation Hierarchy

Description

Function to establish either the Person or the Relation Hierarchy in a multiple network.

Usage

hierar(W, x, type = c("person", "relation"))

Arguments

Details

The set of row inclusions in the Relation Plane of a Relation-Box W defines a partial order: the ego's Relation Hierarchy. Similarly, the ego's or person's inclusions define another partial order: the ego's Person Hierarchy. In other words, the Person Hierarchy captures inclusion relations among actors, whereas the Relation Hierarchy captures inclusion relations among ties. Both hierarchies are defined from the perspective of a selected reference actor in the network.

Notice that the Relation Plane is an outcome from rbox, whereas the cumulative person hierarchy is obtained through the cph function.

Value

A list with an array that represents the partial order structure either of the Person Hierarchy in ph or of the Relation Hierarchy in rh. Both include the reference ego in pers.

References

Breiger, R.L. and P.E. Pattison, “Cumulated social roles: The duality of persons and their algebras,” Social Networks, 8, 215-256. 1986.

See Also

rbox, cph, partial.order, diagram

Examples

# create 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 ) )

# compute person hierarchy of a random actor from relation box
arr |> 
  rbox(k=1) |> 
  hierar(ceiling(runif(1, min=0, max=dim(arr)[2])), type="person")

Incubator networks dataset

Description

These are four data sets collected in the year 2010 (see ‘source’ for the details) of multiple relations between entrepreneurial firms working in business incubators in Denmark.

Each data set contains the adjacency matrices of the three social relations, coded as C, F, and K for working collaboration, informal friendship, and perceived competition among the firms. There are also a pair of actor attributes corresponding to the adoption of two Web innovations in the year 2010 by the firms where A stands for Linkedin and B for Facebook.

In addition, there is a blockmodel attached to each data set that is a product of Compositional Equivalence (cf. cph) with transposes for each type of social tie labelled with the following letter in the Latin alphabet; i.e. D for collaboration, G for friendship, and L for perceived competition.

Usage

data("incubs")
data("incubA")
data("incubB")
data("incubC")
data("incubD")
data("incA")
data("incB")
data("incC")
data("incD")

Format

Each data set is a list with a pair of three-dimensional arrays.

For incubA, the dimensions of net are 26 \times 26 \times 5, and of IM are 4 \times 4 \times 7 (the two attributes led to the identity matrix).

For incubB, the dimensions of net are 18 \times 18 \times 5, and of IM are 4 \times 4 \times 8.

For incubC, the dimensions of net are 22 \times 22 \times 5, and of IM are 3 \times 3 \times 8.

For incubD, the dimensions of net are 15 \times 15 \times 5, and of IM are 4 \times 4 \times 6.

All four networks are together in incubs.

To plot automatically actor attributes in the graph with function multigraph, another version of these data sets are given in incA, incB, incC, and incD, which are "Data.Set" objects class having:

– net for the network data

– atnet a vector that indicates whether or not the arrays in 'net' is attribute data

– IM for the Image Matrices of the reduced network data

– atIM a vector that indicates whether or not the array in 'IM' is attribute data

– cite relational content of the ties

Source

Ostoic, J.A.R. 'Algebraic methods for the analysis of multiple social networks and actors attributes” PhD Thesis. University of Southern Denmark. 2013.

internal function

Description

internal function

internal function

Description

internal function

Construct Multilevel Networks

Description

Function to construct multilevel networks from multimodal structures.

Usage

mlvl(x, y, type = c("bpn", "cn", "cn2"), symCdm, diag, lbs)

Arguments

Details

The inputs in x and y accept multiplex networks and an affiliation network that is high dimensional.

The default multilevel system is a binomial projection bpn that requires data for the two domains, and where the projection is on the system codomain as a directed structure. Making symmetric the codomain structure with symCdm can be convenient for the visualization of multilevel graphs as with mlgraph, and for similar practical reasons, self-relations can be removed from the multilevel structure by disabling diag.

A co-affiliation of network members with option cn2 requires both inputs as well, and the codomain structure in this case is symmetric. However, a common membership network with option cn does not need input in “x” since it returns only the co-affiliation of network members from the codomain structure in y as in cn2.

Since these are different components in the multilevel system for co-affiliation of network members, it is possible to specify the domain and codomain labels in lbs as a list object of length two. Option lbs as a vector can serve instead to label the relation types in the multilevel structure, which may be needed when the codomain in y is just a data frame.

Value

A list object of ‘Multilevel’ class with the chosen type in attribute.

See Also

mlgraph, multigraph

Examples

# array for the domain
arr1 <- round( replace( array(runif(18), c(3,3,2)), array(runif(18), c(3,3,2))>.9, 3 ) )

# rectangle array for the co-domain
arr2 <- round( replace( array(runif(12), c(3,2,2)), array(runif(12), c(3,2,2))>.9, 3 ) )

# multilevel system with default type
mlvl(arr1, arr2)

internal function

Description

internal function

Make Multiple Networks as Monoplex Structures

Description

A function to transform multiple networks into a monoplex structure.

Usage

mnplx(net, directed = TRUE, valued, diag, clu)

Arguments

Details

This function is to collapse multiple types of tie of a network net recorded in an array into a matrix representation with monoplex relations. Other transformations on net are dichotomizing a valued network, and convert directed networks into undirected systems having or not self-relations depending on the value of diag. Moreover, the resulted matrix can be permuted with a clustering information in a vector in clu as with perm.

Value

A matrix of monoplex relations.

See Also

perm, zbind, dichot, reduc

Examples

# create two binary relations among three elements
arr <- round( replace( array(runif(18), c(3,3,2)), array(runif(18),
       c(3,3,2))>.8, 3 ) )

# make array monoplex
mnplx(arr)

internal function

Description

internal function

Actor or Group Neighborhood in Relational System

Description

A function to find with a customized distance the neighbourhood of an actor or group of actors in a relational system representing a simple or multiple network.

Usage

neighb(x, rs, type = c("und", "inn", "out"), k = 1, inclx = FALSE, expand)

Arguments

Details

The relational system in rs represents either the full multiple network of actors or the subset of relational bundles that are mutual or asymmetric. Accordingly, this function identifies the nodes adjacent to ‘x’ within the specified relational system, as well as neighbouring nodes at a user-defined path length. Once the longest path or chain has been reached, increasing k will not add further nodes to the graph. The inn and out options apply to directed networks.

Notice that the output does not differentiate in case the chosen reference actors are in different components of the relational system.

Value

Depending on expand, the output is either a vector or a list with the neighbour nodes to the reference actor(s).

See Also

expos, rel.sys, bundles

Examples

# create two binary relations among three elements
arr <- round( replace( array( runif(18), c(3 ,3, 2) ), array( runif(18),
       c(3, 3, 2) ) > .9, 3 ) )

# establish the system of strong bonds
rs <- rel.sys(arr, bonds = "strong")

# obtain inmediate neighbourhood of the first node
neighb(1, rs)

internal function

Description

internal function

Import Log-Report from Pacnet

Description

A function to read output files and import from the Pacnet program with full factorization option.

Usage

pacnet(file, toarray = FALSE, uniq = FALSE, transp = FALSE, sep)

Arguments

Details

This function is used to read the output file from the Pacnet program, which typically has the .out extension. By default the result is given in a list format, but it is possible to transform the pair lists into arrays.

Note that the options in the Pacnet program should include the full factorization in the output; otherwise the object will be NULL. Also note that only partial order structures of order 36 and less are currently supported.

Value

A ‘Pacnet’ class object with items:

References

Pattison, P, S Wasserman, G Robins and AM Kanfer “Statistical Evaluation of Algebraic Constraints for Social Networks,” Journal of Mathematical Psychology, 44, 536-568. 2000

See Also

pi.rels, cngr, decomp, write.dat

Partial Order of String Relations or of Galois Derivations

Description

Construct the partial order table of unique relations of the semigroup, or else of the concepts produced by Galois derivations.

Usage

partial.order(x, type = c("strings", "galois", "pi.rels"), lbs, sel, po.incl, valued)

Arguments

Details

To get the partial order of an entire semigroup, both generators and compound relations must be considered. This information and the labels of the unique relations are given by the strings function. cf. semigroup to see how the x should be specified properly.

Galois derivations are now possible to be partially ordered as well, and this option is based on the output given by the galois function.

Value

An object of ‘Partial.Order’ class with the partial order table in a matrix format.

References

Pattison, P.E. Algebraic Models for Social Networks. Cambridge University Press. 1993.
Ganter, B. and R. Wille Formal Concept Analysis – Mathematical Foundations. Springer. 1996.

See Also

as.strings, strings, galois, perm, diagram, fltr.

Examples

# load a dataset 
data("incubA")

# strings in structure and partial order
strings(incubA$IM) |> 
  partial.order()

Array Permutation

Description

Function to permutate a given array of relations.

Usage

perm(x, clu, rev, lbs, sort)

Arguments

Details

This function performs a permutation of a given array or matrix representing relations according to a clustering vector of membership.

Value

A permuted matrix or array

See Also

cph, partial.order

Examples

# scan the multiplication table data
s <- matrix(data=c(1, 1, 1, 3, 3, 3, 3, 3, 3), nrow=3, ncol=3, byrow=TRUE)

# the permutation as an endomorphism
perm(s, clu = c(1,2,3))

Pathfinder Valued Networks and Triangle Inequality

Description

A function to produce the skeleton of a valued network with the pathfinder algorithm and triangle inequality.

Usage

pfvn(x, r, q)

Arguments

Details

Pathfinder analysis is based on a symmetric adjacency matrix or array in x representing valued networks where the values reflect the “proximity” between pairs of network members. This function depends heavily on internal function cpath where the aim is to produce the skeleton or the salient structure of x. The pathfinder structure is for undirected networks, whereas for directed network structures the triangle inequality principle is applied.

Parameter q represents the length of all walks computed over the semiring in the produced dissimilarity matrix, while r is a distance measure computed by the Minkowski operation.

Value

References

Schvaneveldt, R., Durso, F. and Dearholt, D., “Network structures in proximity data,” in G. Bower, ed., The Psychology of Learning and Motivation: Advances in Research & Theory, vol. 24, Academic Press, pp. 249-284. 1989.
Batagelj, V., Doreian, P., Ferligoj, A. and Kejzar, N., Understanding Large Temporal Networks and Spatial Networks: Exploration, Pattern Searching, Visualization and Network Evolution, Wiley. 2014.

See Also

galois, cscl, multigraph.

Examples

# create valued network data
arr <- round( array(runif(18), c(3,3,2)), array(runif(18), c(3,3,2)) ) * 10L

# pathfinder valued network of 'arr'
## Not run: 
pfvn(arr)

## End(Not run)

\pi-Relations

Description

A function to produce the \pi-relations of a partially ordered structure.

Usage

pi.rels(x, po.incl, vc, po)

Arguments

Details

The totality of \pi-relations are represented with an array format, and the arrangement of meet-complements in this array are congruences that serve for the decomposition of a partially ordered semigroup structure.

Induced inclusions to the partial order in x as a “Ind.incl” class object are produced by the fact function, while induced inclusions as a “Pacnet” class object comes from the outcome of a Pacnet log-report. With either case, \pi-relations are derived from the induced inclusions in a factorization process of a partially ordered semigroup.

A visualization of the \pi-relations partial order arrangement with function diagram is feasible provided that the resulted “Pi.rels” class object computes a “Pi.rels” class with the type option "pi.rels" in function partial.order. In such case, including the partial order with po.incl and a addition in po is a good idea since the partial order is the minimal element of the lattice of congruence relations.

Producing the partial order structure of a set of \pi-relations can be resource consuming and for exploring and other purposes, with vc option one can limit the amount of induced inclusions to compute.

Value

An object of the “Pi.rels” class with:

References

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

See Also

fact, decomp, partial.order, diagram, semigroup, pacnet

internal function

Description

internal function

internal function

Description

internal function

Construct the Relation-Box of Multiple Network

Description

Function to construct the Relation-Box of a multiple network.

Usage

rbox(w, transp = FALSE, smpl = FALSE, k = 3, tlbs)

Arguments

Details

If transp = TRUE the labels of the transpose are toggle case of the labels of the original matrices, and in such case, it is advised to simplify the strings of relations. To prevent a transposed structure for a certain array of w, use NA in the vector the transpose labels tlbs corresponding to the respective matrix.

Values of k until 9 are supported.

Value

An object of the ‘Rel.Box’ class.

Warning

May take a long time of computation with many types of relations and when the order of the multiplex network is high, and turning k to more than three.

References

Winship, C. and M.J. Mandel “Roles and positions: A critique and extension of the blockmodelling approach,” Sociological Methodology, 314-344. 1983.

See Also

cph, semigroup, hierar

Examples

# load the data
data("incubA")

# relation box of image matrices in dataset
## Not run: 
incubA |> getElement("IM") |> 
  rbox()

## End(Not run)

Read dl Files

Description

A function to read files with the Ucinet dl format.

Usage

read.dl(file)

Arguments

Details

Ucinet dl format is to represent multiple and two-mode network structures. It is used in NetDraw, which is a component of the Ucinet program. Besides multiple networks, this function can read two-mode structures as well but the ‘EDGELIST’ option in DL is not yet supported for reading.

Value

A data frame for two-mode networks, or an array representing the multiple networks with one set of actors.

References

Borgatti, S.P., NetDraw Software for Network Visualization. Analytic Technologies. 2002.
Borgatti, S.P., Everett, M.G. and Freeman, L.C. Ucinet for Windows: Software for Social Network Analysis. Analytic Technologies. 2002.

See Also

write.dl, edgel, read.gml

Read gml Files

Description

A function to read files with the gml format.

Usage

read.gml(file, as = c("edgel", "array"), directed = TRUE, coords = FALSE)

Arguments

Details

The gml format, an acronym for graph modelling language, provides capabilities to represent multiple networks and add arguments both to the nodes and the edges for visualization purposes.

For the multiplexity in the ties the gml file distinguishes “graphics” arguments inside “edge”. Both “style” and “fill” are supported here and the former has priority over the latter in case the two are given; otherwise when these arguments are absent, the function separates up to a couple of relational levels when several pairwise ties are specified.

Node attributes can be retrieved as well when coordinates in coords are chosen.

Value

Depending the option chosen, the output is either a data frame or an array representing the multigraph. If the coordenates are chosen then these are part of the obejct structure, but they are not visible.

References

visone: Software for the analysis and visualization of social networks. http://visone.info

See Also

write.gml, edgel, read.dl

internal function

Description

internal function

Reduce Matrices or Arrays

Description

Function to reduce a matrix or an array with a given clustering vector.

Usage

reduc(x, clu, lbs, slbs, valued, row, col)

Arguments

Details

Given a partition, this function serves to reduce either a matrix representing e.g. a partial order structure. However, the reduction is also generalized to three-dimensional arrays representing multiple relations or affiliation networks with either row or col.

Notice that the “reduction” of a semigroup of x is made through a decomposition process that is computed with function decomp.

Value

A reduced matrix or a reduced three-dimensional array of the input data according to the clustering information.

See Also

cngr, rbox, decomp

Examples

# scan the multiplication table data
s <- matrix(data=c(1, 1, 1, 3, 3, 3, 3, 3, 3), nrow=3, ncol=3, byrow=TRUE)

# reduce the multiplication table
s |> reduc(clu=c(1,2,2))

internal function

Description

internal function

Relational System of Network

Description

Function to create a Relational System of a multiple network.

Usage

rel.sys(x, type = c("tolist", "toarray"), bonds = c("entire", "strong", "weak", 
        "asym", "recp", "txch", "tent", "mixd", "full"), loops = FALSE, 
        sel = NULL, att = NULL, sep)

Arguments

Details

A relational system of network x is a configuration with a particular pattern like the bond type among members. For instance, when the type of bonds chosen is entire then the nodes with ties are considered in the relational system without isolated nodes. Besides, strong bonds are relational bundles with a mutual character, whereas weak bonds are those patterns exclusively without mutual character.

It is also possible to select the network members having or *not* having the attribute that is specified in the Attrs output by using in argument sel. This is possible when choosing from a list with actor attributes.

Value

For type = "tolist" (default) option, an object of ‘Rel.System’ class where items are:

For type = "toarray", the output is a two or three dimensional dichotomous array recording the relations among the actors in the network.

References

Ostoic, J.A.R. “Creating context for social influence processes in multiplex networks.” Network Science, 5(1), 1-29.

See Also

expos, bundles, neighb

Examples

# create two binary relations among three elements
arr <- round( replace( array( runif(18), c(3 ,3, 2) ), array( runif(18),
       c(3, 3, 2) ) > .9, 3 ) )

# establish system of strong bonds
arr |> rel.sys(bonds = "strong")

# first array is for attributes
arr |> rel.sys(att = 1)

# select first node
arr |> rel.sys(sel = 1)

Remove Isolates from Network

Description

Function to remove isolate nodes in simple and multiple networks.

Usage

rm.isol(x, diag, diag.incl)

Arguments

Details

Isolated nodes do not have any edges in the network, and in a multivariate system, there is no edges adjacent to these kinds of nodes at any level.

Value

A matrix or an array representing a multiple network without the isolated actors.

See Also

edgel, zbind

Examples

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

# remove isolates (if exist)
rm.isol(arr)

Semigroup of Relations for 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)

Arguments

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. For other values 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, and depend on function cscl.

Value

A ‘Semigroup’ class object with items:

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.

Warning

Computing for the semigroup construction can take very long time for medium size or bigger sets (e.g. having more than 4 relation types).

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.

See Also

green.rel, strings, edgeT, wordT, cngr, cscl

Examples

# create two binary relations among three elements
arr <- round( replace( array( runif(18), c(3 ,3, 2) ), array( runif(18),
       c(3, 3, 2) ) > .9, 3 ) )

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

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

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, k = 2, lbs)

Arguments

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 determine whether a given signed network is balanced or clusterable. The symmetric closure evaluates the assessment by looking at semicycles in the system; otherwise, the evaluation is through closed paths.

Value

A ‘Semiring’ class object with items:

Warning

Disabling transitive closure should be made with good substantial reasons.

References

Harary, F, Z. Norman, and D. Cartwright Structural Models: An Introduction to the Theory of Directed Graphs. Wiley. 1965.
Doreian, P., V. Batagelj and A. Ferligoj Generalized Blockmodeling. Cambridge University Press. 2004.
Ostoic, J.A.R. “Creating context for social influence processes in multiplex networks.” Network Science, 5(1), 1-29.

See Also

signed, as.signed

Examples

# create two binary relations among three elements
arr <- round( replace( array( runif(18), c(3 ,3, 2) ), array( runif(18),
       c(3, 3, 2) ) > .9, 3 ) )

# create signed matrix with the two types of relations
# and establish the semiring structure
signed(arr) |> 
  semiring()

Signed Network

Description

Construct the signed network of a system made of contrasting relations.

Usage

signed(P, N = NULL, lbs)

Arguments

Details

This function coerces an array(s) to become a ‘Signed’ object. Positive ties are always in the first argument, and in case that this array has three dimensions, the second dimension is considered as the negative ties, provided that N is still NULL.

If ambivalent ties are present in the structure then the signed matrix represent positive, negative, ambivalent, and null ties as p, n, a, and o respectively; otherwise, the values are 1, -1, and 0.

Value

A ‘Signed’ class object with items:

A warning message is shown when the N argument has more than two dimensions.

References

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

See Also

semiring, as.signed

Examples

# load a dataset
data("incubA")

# make signed matrix of image matrix in dataset
incubA |> getElement("IM") |> signed()

internal function

Description

internal function

Strings of Relations

Description

Function to obtain labels of unique relations, and generators and compound relations of a complete semigroup structure.

Usage

strings(x, equat = FALSE, k = 2, smpl, valued)

Arguments

Details

The strings are the unique relations, which constitute the elements of the complete semigroup. These are both the generators and the compound relations after applying the Axiom of Quality, which means that even some generators can be disregarded.

This function is especially useful to construct the partial order of relations and to establish the set of equations in the relational structure.

The maximum length of the strings in the equations is currently 4.

Value

An object of ‘Strings’ class.

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.

See Also

partial.order, semigroup.

Examples

# create two binary relations among three elements
arr <- round( replace( array( runif(18), c(3 ,3, 2) ), array( runif(18),
       c(3, 3, 2) ) > .9, 3 ) )

# get string relations
strings(arr)

internal function

Description

internal function

Summary of Bundle Classes

Description

Function for pretty printing of bundle class patterns results.

Usage

summaryBundles(x, file = NULL, latex = FALSE, byties)

Arguments

Details

This function prints the bundle census patterns existing in the network with an option to export such information in a friendly format. The dyadic bundle patterns are provided by the function bundles; however, the outcome of this function provides a list of pair lists for each bundle with the involved types of relations and nodes in the network. This form for presentation, although is convenient for further computation, it is not always easy to read for the human eye. The pair separator used to print the bundle occurrences is taken from the output of the bundles function.

If latex is set to TRUE, then the path file is activated to obtain a tex file with the different bundle class patterns. Finally, the optional argument byties provide more precise information about the patterned ties disregarding the relational content.

Value

The distinct bundle class patterns with a user friendly format.

Warning

Files with identical path and name to file will be overwritten.

References

Ostoic, J. A. R. “Dyadic Patterns in Multiple Networks,” Advances in Social Networks Analysis and Mining, International Conference on, 475-481. 2011.

See Also

bundles, bundle.census

Examples

# create two binary relations among three elements
arr <- round( replace( array( runif(18), c(3 ,3, 2) ), array( runif(18),
       c(3, 3, 2) ) > .9, 3 ) )

# print different relational bundles in 'arr'
arr |> 
  bundles() |> 
  summaryBundles()

internal function

Description

internal function

internal function

Description

internal function

Transform Data from-to Matrix-List Formats

Description

Function to transform data from/to matrix/list formats or edge list representing a network.

Usage

transf(x, type = c("toarray", "tolist", "toarray2", "toedgel"), lbs = NULL, lb2lb, valued, 
       sep, ord, sort, sym, add, adc, rm.isol, na.rm)

Arguments

Details

Option "tolist" is for transforming a matrix or an array to a list of pair elements. In case that the lb2lb is enabled in this type of transformation, then lbs must be provided, whereas the pair separator is optional. Notice that the default of lb2lb is TRUE for "toarray", while is FALSE for "tolist".

Option type "toarray" will produce a matrix from a list of pair elements, and in this case is advisable to specify the order of the structure. Three dimensional structures are supported in the transformations with all options.

Data frames are also accepted for the "tolist" option; however, in case that this information is given as a list of pair relations the output will be a square matrix.

When the transformation option is "edgel", the output is a data frame with the first two columns for the sending and receiving ties. For simple networks, these two columns are enough and for multiplex networks additional columns are for the types of tie, one for each (cf. function edgel).

Notice that for high dimensional arrays, the rel.sys function provides additional information other than the list of pair relations of the entire structure.

Value

Depending on the input data, the result is either a list of pair relations or a matrix of relations.

See Also

edgel, bundles, reduc, rel.sys

Examples

# scan the multiplication table data
s <- matrix(data=c(1, 1, 1, 3, 3, 3, 3, 3, 3), nrow=3, ncol=3, byrow=TRUE)

# transform the matrix to a list format
s |> transf(lb2lb = TRUE, lbs = c("n","m","u"))

internal function

Description

internal function

internal function

Description

internal function

internal function

Description

internal function

Word Table of Relations

Description

Function to produce the Word Table of multiple relations as representation form of a semigroup of relations.

Usage

wordT(x)

Arguments

Details

The Word Table is a consequence of the Edge Table (cf. edgeT) and the function gives a list of indexed elements in the complete semigroup.

In terms of the Cayley graph of the semigroup (cf. ccgraph, the collection of unique relations (both compound and generators) are represented by nodes. On the other hand, the generators are edges that record the result of post-multiplying the compound relations by the generators.

The labels for the elements can be retrieved by the strings function.

The generators do not have values in neither the “node” nor the “generator” of the Word table since they are not a product of any other element in the semigroup (cf. ‘details’ for the rest of the values).

Value

A ‘WordTable’ class object with items:

References

Cannon, J.J. “Computing the ideal structure of finite semigroup,” Numerische Mathematik, 18, 254-266. 1971.
Pattison, P.E. Algebraic Models for Social Networks. Cambridge University Press. 1993.

See Also

edgeT, semigroup, strings, ccgraph

Examples

# create two binary relations among three elements
arr <- round( replace( array( runif(18), c(3 ,3, 2) ), array( runif(18),
       c(3, 3, 2) ) > .9, 3 ) )

# obtain word table
wordT(arr)

Write dat Files

Description

A function to write dat files.

Usage

write.dat(x, path)

Arguments

Details

“dat” files are the format used in the Pacnet program inside StOCNET. In case that the input data represents a multiple network then a separate file will be produced, each file representing a single type of relationship in the system. The name of the output files depends on the object title.

Value

File(s) output with adjacency matrices with a .dat format.

In case that a directory in path for output does not exist then the folder is created.

References

StOCNET An open software system for the advanced statistical analysis of social networks.

See Also

pacnet, write.gml, write.dl

Write dl Format Files

Description

A function to write dl files representing multiple networks.

Usage

write.dl(x, file = NULL, type = c("nodelist", "fullmat"))

Arguments

Details

Files with dl format serve to represent multiple networks, and it is one used in NetDraw, which is a component of the Ucinet program.

Value

A file in path with the data with a .dl format.

References

Borgatti, S.P., NetDraw Software for Network Visualization. Analytic Technologies. 2002.
Borgatti, S.P., Everett, M.G. and Freeman, L.C. Ucinet for Windows: Software for Social Network Analysis. Analytic Technologies. 2002.

See Also

read.dl, write.gml, write.edgel, write.dat

Write edge-list files

Description

A function to write edge-list files having columns for sender, receiver, and ties in multiplex networks.

Usage

write.edgel(x, file = NULL, sep = "\t", header = TRUE)

Arguments

Details

Write edge-list files with a send, receive, and ties, which is a data frame with at least two columns for the sender and receiver, and the different types of tie for multiplex networks, one column for each type of relation.

Value

A data frame having an edge-list format in a file with path or an object.

See Also

edgel, , write.dl

Write gml Files

Description

A function to write files with a graph modelling language (gml) format.

Usage

write.gml(x, file = NULL)

Arguments

Details

The gml format, an acronym for graph modelling language, provides capabilities to represent multiple networks and add arguments to both the nodes and the edges for visualization purposes.

Value

A graph with modelling language format in a file with path or an object.

Warning

Files with identical path and name to file will be overwritten.

See Also

read.gml, write.dl, write.dat

Bind Matrices and Multidimensional Arrays

Description

A function to combine or bind matrices and multidimensional arrays.

Usage

zbind(..., sort, force)

Arguments

Details

Function zbind is for stacking for instance two-dimensional arrays into a single three-dimensional object to represent a multivariate system structure. Both square and rectangular arrays are supported provided that the dimensions in the input are equal, and data frames should be transformed into arrays for the binding.

The dimnames in the array output correspond to the names of the first array in the input, and a warning message is given when the dimnames are NULL.

Argument force must be set to TRUE for matrices and arrays with different dimensions, and a message is given.

Value

Mostly a three dimensional array.

Warning

Arrays without dimnames are not fully supported.

See Also

mnplx, dichot, strings

Examples

# create two sets with two binary relations among three elements
arr1 <- round( replace( array( runif(18), c(3 ,3, 2) ), array( runif(18),
        c(3, 3, 2) ) > .9, 3 ) )

arr2 <- round( replace( array( runif(18), c(3 ,3, 2) ), array( runif(18),
        c(3, 3, 2) ) > .5, 3 ) )

# bind sets of data
zbind(arr1, arr2)

internal function

Description

internal function