Global triad closure from a triad census

Given a triad census of a suitable scheme, calculate a global measure of triad closure for the associated affiliation network.

triad_closure_from_census(census, scheme = NULL, alcove = 0, wedge = 0,
  maps = 0, congruence = 0, measure = NULL, open.fun = NULL,
  closed.fun = NULL, counts = FALSE)

triad_closure_from_simple_census(census, alcove = 0, wedge = 0, maps = 0,
  congruence = 0, open.fun = NULL, closed.fun = NULL, counts = FALSE)

triad_closure_from_binary_census(census, alcove = 0, wedge = 0, maps = 0,
  congruence = 0, open.fun = NULL, closed.fun = NULL, counts = FALSE)

triad_closure_from_difference_census(census, alcove = 0, wedge = 0,
  maps = 0, congruence = 0, open.fun = NULL, closed.fun = NULL,
  counts = FALSE)

triad_closure_from_full_census(census, alcove = 0, wedge = 0, maps = 0,
  congruence = 0, open.fun = NULL, closed.fun = NULL, counts = FALSE)

wedges_from_full_census(census, open.fun, closed.fun)

wedges_from_census(...)

wedgecount_census(...)

wedgecount.census(...)

triad_closure_from_census_original(census, scheme = NULL, alcove = 0,
  wedge = 0, maps = 0, congruence = 0, measure, open.fun, closed.fun,
  counts = FALSE)

transitivity_from_census(...)

transitivity.census(...)

Arguments

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`; otherwise it is treated as full.
scheme	Character; the type of triad census provided, matched to `"full"`, `"difference"` (also `"uniformity"`), `"binary"` (also `"structural"`), or `"simple"`.
alcove, wedge, maps, congruence	Choice of alcove, wedge, maps, and congruence (see Details).
measure	Character; the measure of triad closure (matched to "classical", "watts_strogatz", "twomode", "opsahl", "unconnected", "liebig_rao_0", "completely_connected", "liebig_rao_3", "exclusive", "allact", "indequ", "indstr", "injact", "injequ", or "injstr"). Overrides `alcove`, `wedge`, `maps`, and `congruence`.
open.fun, closed.fun	Functions to calculate the open and closed wedge count for a triad (when `scheme` is `"full"`) or a triad census (otherwise), in order to calculate a custom measure of triad closure. Override `measure`.
counts	Logical; whether to return open and closed wedge counts instead of the quotient.
...	Arguments passed from deprecated functions to their replacements.

Details

Each global measure of triad closure can be recovered from the full triad census, and some can be recovered from smaller censuses. This function verifies that a given census is sufficient to recover a given measure of triad closure and, if it is, returns its value.

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.

Measures of triad closure

Each measure of triad closure is defined as the proportion of wedges that are closed, where a wedge is the image of a specified two-event triad \(W\) under a specified subcategory of graph maps \(C\) subject to a specified congruence relation \(~\), and where a wedge is closed if it is the image of such a map that factors through a canonical inclusion of \(W\) to a specified self-dual three-event triad \(X\).

The alcove, wedge, maps, and congruence can be specified by numerical codes as follows (no plans exist to implement more measures than these):

alcove:
- 0: \(T_{(1,1,1),0}\)
- 1: \(T_{(1,1,0),1}\) (not yet implemented)
- 2: \(T_{(1,0,0),2}\) (not yet implemented)
- 3: \(T_{(0,0,0),3}\) (not yet implemented)
wedge:
- 0: \(T_{(1,1,0),0}\)
- 1: \(T_{(1,0,0),1}\) (not yet implemented)
- 2: \(T_{(0,0,0),2}\) (not yet implemented)
maps:
- 0: all graph maps (injective on actors)
- 1: injective graph maps
- 2: induced injective graph maps
congruence:
- 0: same actor and event images (equivalence)
- 1: same actor images, structurally equivalent event images
- 2: same actor images

Some specifications correspond to statistics of especial interest:

0,0,0,2: the classical clustering coefficient (Watts & Strogatz, 1998), evaluated on the unipartite actor projection
0,0,1,0: the two-mode clustering coefficient (Opsahl, 2013)
0,0,2,0: the unconnected clustering coefficient (Liebig & Rao, 2014)
3,2,2,0: the completely connected clustering coefficient (Liebig & Rao, 2014) (not yet implemented)
0,0,2,1: the exclusive clustering coefficient (Brunson, 2015)
0,0,2,2: the exclusive clustering coefficient

See Brunson (2015) for a general definition and the aforecited references for discussions of each statistic.

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.

Watts, D.J., & Strogatz, S.H. (1998). Collective dynamics of "small-world" networks. Nature, 393(6684), 440--442.

Opsahl, T. (2013). Triadic closure in two-mode networks: Redefining the global and local clustering coefficients. Social Networks, 35(2), 159--167. Special Issue on Advances in Two-mode Social Networks.

Liebig, J., & Rao, A. (2014). Identifying influential nodes in bipartite networks using the clustering coefficient. Pages 323--330 of: Proceedings of the tenth international conference on signal-image technology and internet-based systems.