Description Details Triad censuses Measures of triad closure Author(s) References

Calculate triad censuses and triad closure statistics designed for affiliation networks.

The package contains two principal tools for the triadic analysis of affiliation networks: triad censuses and measures of triad closure. Assorted additional functions, including a measure of dynamic triad closure, are also included.

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,*λ=(λ_1≤qλ_2≤qλ_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*λ*and columns indexed by*w*, with the partitions*λ*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

*λ=(x≥ y≥ z)*and*w*above, note that the numbers of shared events for each pair and for the triad are*x+w≥ y+w≥ z+w≥ w≥ 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.

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.

Jason Cory Brunson

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*.

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

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