bitriad: Triadic analysis of affiliation networksCalculate 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,
\(\lambda=(\lambda_1\leq\lambda_2\leq\lambda_3)\), 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.
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.
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.