Triad census for affiliation networks

Given an affiliation network, tally all actor triads by isomorphism or other congruence class.

Usage

triad_census(graph, ..., add.names = TRUE)

triad_census_an(
  graph,
  scheme = "full",
  method = "batagelj_mrvar",
  ...,
  add.names = TRUE
)

triad.census.an(...)

triad_census_full(graph, method = "batagelj_mrvar", ..., add.names = TRUE)

triad_census_full_batagelj_mrvar(graph, use.integer = FALSE)

triad_census_full_projection(graph, verbose = FALSE)

triad_census_difference(
  graph,
  method = "batagelj_mrvar",
  ...,
  add.names = TRUE
)

triad_census_difference_batagelj_mrvar(graph, use.integer = FALSE)

triad_census_difference_projection(graph)

unif_triad_census(graph)

unif.triad.census(graph)

triad_census_binary(graph, method = "batagelj_mrvar", ..., add.names = TRUE)

triad_census_binary_batagelj_mrvar(graph, use.integer = FALSE)

triad_census_binary_projection(graph, verbose = FALSE)

str_triad_census(graph)

structural.triad.census(graph)

simple_triad_census(graph, add.names = TRUE)

simple.triad.census(graph, add.names = TRUE)

Arguments

graph: An igraph object, usually an affiliation network.
...: Additional arguments (currently use.integer and verbose) passed to the method function.
add.names: Logical; whether to label the rows and columns of the output matrix.
scheme: Character; the type of triad census to calculate, matched to "full", "difference" (also "uniformity"), "binary" (also "structural"), or "simple".
method: Character; the triad census method to use. Currently only "batagelj_mrvar" is implemented. "projection" calls an inefficient but reliable implementation in R from the first package version that invokes the simple_triad_census() of the actor_projection() of graph.
use.integer: Logical; whether to use the IntegerMatrix class in Rcpp rather than the default NumericMatrix.
verbose: Logical; whether to display progress bars.

Value

An integer matrix of counts of triad congruence classes, with row indices reflecting pairwise exclusive events and column indices reflecting triadwise events.

Details

The triad_census_*() functions implement the several triad censuses described below. Each census is based on a congruence relation among the triads in an affiliation network, and each function returns a matrix (or, in the "simple" case, a vector) recording the number of triads in each congruence class.

The function triad_census() masks triad_census() but calls it in case graph is not an affiliation network.

Triad censuses

Three triad censuses are implemented for affiliation networks:

The full triad census (Brunson, 2015) records the number of triads of each isomorphism class. The classes are indexed by a partition, \(\lambda=(\lambda_1\leq\lambda_2\leq\lambda_3)\), indicating the number of events attended by both actors in each pair but not the third, and a positive integer, \(w\), indicating the number of events attended by all three actors. The isomorphism classes are organized into a matrix with rows indexed by \(\lambda\) and columns indexed by \(w\), with the partitions \(\lambda\) ordered according to the revolving door ordering (Kreher & Stinson, 1999). The main function triad_census_an (called from triad_census when the graph argument is an affiliation_network) defaults to this census.
For the analysis of sparse affiliation networks, the full triad census may be less useful than information on whether the extent of connectivity through co-attended events differs between each pair of actors. In order to summarize this information, a coarser triad census can be conducted on classes of triads based on the following congruence relation: Using the indices \(\lambda=(x\ge y\ge z)\) and \(w\) above, note that the numbers of shared events for each pair and for the triad are \(x+w\ge y+w\ge z+w\ge w\ge 0\). Consider two triads congruent if the same subset of these weak inequalities are strictly satisfied. The resulting difference triad census, previously called the uniformity triad census, implemented as triad_census_difference, is organized into a \(8\times 2\) matrix with the strictness of the first three inequalities determining the row and that of the last inequality determining the column.
A still coarser congruence relation can be used to tally how many are connected by at least one event in each distinct way. This relation considers two triads congruent if each corresponding pair of actors both attended or did not attend at least one event not attended by the third, and if the corresponding triads both attended or did not attend at least one event together. The binary triad census (Brunson, 2015; therein called the structural triad census), implemented as triad_census_binary, records the number of triads in each congruence class.
The simple triad census is the 4-entry triad census on a traditional (non-affiliation) network indicating the number of triads of each isomorphism class, namely whether the triad contains zero, one, two, or three links. The function simple_triad_census computes the classical (undirected) triad census for an undirected traditional network, or for the actor projection of an affiliation network (if provided), using triad_census; if the result doesn't make sense (i.e., the sum of the entries is not the number of triples of nodes), then it instead uses its own, much slower method.

Each of these censuses can be projected from the previous using the function project_census. A fourth census, called the uniformity triad census and implemented as unif_triad_census, is deprecated. Three-actor triad affiliation networks can be constructed and plotted using the triad functions.

The default method for the two affiliation network–specific triad censuses is adapted from the algorithm of Batagelj and Mrvar (2001) for calculating the classical triad census for a directed graph.

References

Kreher, D.L., & Stinson, D.R. (1999). Combinatorial algorithms: generation, enumeration, and search. SIGACT News, 30(1), 33–35.

Batagelj, V., & Mrvar, A. (2001). A subquadratic triad census algorithm for large sparse networks with small maximum degree. Social Networks, 23(3), 237–243.

Brunson, J.C. (2015). Triadic analysis of affiliation networks. Network Science, 3(4), 480–508.

Examples

data(women_clique)
(tc <- triad_census(women_clique, add.names = TRUE))
#>         0 1
#> (0,0,0) 0 0
#> (1,0,0) 0 1
#> (1,1,0) 0 3
#> (1,1,1) 1 0
#> (2,0,0) 0 0
#> (2,1,0) 3 0
#> (2,1,1) 2 0
#> (2,2,0) 0 0
#> (2,2,1) 0 0
#> (2,2,2) 0 0
sum(tc) == choose(vcount(actor_projection(women_clique)), 3)
#> [1] TRUE