Given a triad census of any scheme, construct a triad census of a coarser (strictly less informative) scheme.
project_census(census, scheme = NULL, add.names = TRUE) project.census(census, scheme = NULL, add.names = TRUE) difference_from_full_census(census) ftc2utc(census) binary_from_full_census(census) ftc2stc(census) simple_from_full_census(census) ftc2tc(census) binary_from_difference_census(census) utc2stc(census) simple_from_difference_census(census) utc2tc(census) simple_from_binary_census(census) stc2tc(census)
| census | Numeric matrix or vector; an affiliation network triad census.
It is treated as binary or simple if its dimensons are 4-by-2 or 4-by-1,
respectively, unless otherwise specified by |
|---|---|
| scheme | Character; the type of triad census provided, matched to
|
| add.names | Logical; whether to label the rows and columns of the output matrix. |
This function inputes an affiliation network triad census of any
scheme and returns a list of triad censuses projected from it (not icluding
itself). The schemes are, in order of resolution, full (also called
the affiliation network triad census without qualification),
difference, binary, and simple. A final element of the
output list is the total number of triads in the affiliation network. Each
summary can be recovered from those before it, specifically by aggregating
certain matrix entries to form a smaller matrix. The helper functions
*_from_*_census() project a census of each scheme to one of each
coarser scheme.
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.
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.
Other triad census functions: triad_census,
triad_closure_from_census,
triad_tallies