Package {bitriad}

bitriad: Triadic analysis of affiliation networks

Description

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

Details

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.

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.

Author(s)

Jason Cory Brunson

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.

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

See Also

Useful links:

• http://corybrunson.github.io/bitriad/

Affiliation network structure

Description

Test igraph objects for affiliation network structure or impose such structure if possible.

Usage

is_an(graph)

is.an(graph)

as_an(graph, map_type = FALSE)

as.an(graph, map_type = FALSE)

Arguments

Details

An affiliation network is a bipartite graph whose nodes are classified as actors and events in such a way that all links are between actors and events. The function is_bipartite() tests an igraph object for a type attribute, which is intended to bipartition of the nodes. It does not test whether the links respect this partition. The function is_an tests this, as well as the condition that actor nodes precede event nodes in their node IDs, which simplifies some other functions. The function as_an coerces an igraph object to an affiliation network by verifying that the object is bipartite and minimally permuting the node IDs. If graph has no type attribute and map_type is FALSE, then as_an throws an error; if map_type is TRUE, then as_an calls bipartite_mapping() to add a logical type attribute that takes the value FALSE at the first node (by node ID) in each connected component and TRUE or FALSE at the remaining nodes. (If this cannot be done, an error is thrown.)

Value

For is_an(), a logical value; for as_an(), the input graph with a "type" attribute.

See Also

Original igraph functions: is_igraph()

Other network testing and coercion: dynamic_an

Examples

graph <- make_graph(c( 1,2, 1,4, 1,6, 3,6, 5,6, 7,8, 9,8 ))
is_an(graph)
# as_an(graph) # throws an error
an_graph <- as_an(graph, map_type = TRUE)
is_an(an_graph)

Corporate interlocks among directors in Chicago.

Description

An affiliation network of 20 directors of 24 corporations and social clubs.

Format

An affiliation network; see is_an.

Source

Barnes, R. & Burkett, T. (2010). Structural Redundancy and Multiplicity in Corporate Networks. Connections, 30(2), p. 4–20. https://www.insna.org/connections-archives-2000-2015

Take the dual of an affiliation network

Description

This function obtains the dual of an affiliation network, in which the actors and events have swapped roles. To do this, it negates the logical values of the node type attribute and reorders the node ids accordingly.

Usage

dualize(graph)

dual_an(graph)

dual.an(graph)

Arguments

Value

The input graph with the "type" attribute logically negated.

See Also

Other modal queries and manipulations: mode_addition, mode_counts, modes, schedule()

Examples

data(women_clique)
( tab <- table(V(women_clique)$type) )
( proj <- actor_projection(dualize(women_clique)) )
vcount(proj) == tab[2]

Dynamic affiliation network structure

Description

An affiliation network is dynamic, for present purposes, if its event nodes have time stamps, recorded as a numeric vertex attribute "time" in non-decreasing order.

Usage

is_dynamic_an(graph)

is.dyn(graph)

as_dynamic_an(graph, use_attr = NULL)

Arguments

Value

The input graph with a new "time" vertex attribute and its event nodes permuted to respect this attribute.

See Also

Other network testing and coercion: affiliation_network

Examples

data(women_group)
is_dynamic_an(women_group)
data(women_clique)
is_dynamic_an(women_clique)
as_dynamic_an(women_clique)
data(nmt_meetings)
nmt_meetings <- set_event_attr(
  nmt_meetings, "meet",
  value = gsub("meet", "", V2(nmt_meetings)$name)
)
as_dynamic_an(nmt_meetings, use_attr = "meet")

Triadic closure for dynamic affiliation networks

Description

Given an affiliation network with time-stamped events, compute the proportion of centered triples at which an open wedge exists at some time that is closed at a later time.

Usage

dynamic_triad_closure(
  graph,
  actors = V(graph)[V(graph)$type == FALSE],
  type = "global",
  ...,
  measure = NULL
)

dynamic_triad_closure_an(
  graph,
  actors = V(graph)[V(graph)$type == FALSE],
  type = "global",
  ...,
  measure = NULL
)

dynamic_transitivity_an(
  graph,
  actors = V(graph)[V(graph)$type == FALSE],
  type = "global",
  ...,
  measure = NULL
)

dyn.transitivity.an(
  graph,
  actors = V(graph)[V(graph)$type == FALSE],
  type = "global",
  ...,
  measure = NULL
)

dynamic_triad_closure_projection(graph, memory = Inf, type = "global")

Arguments

Value

A numeric vector of the same length as actors.

See Also

Other triad closure functions: project_transitivity(), transitivity_an(), triad_closure(), triad_closure_from_census()

Examples

data(women_group)
dynamic_triad_closure(women_group)
cbind(
  transitivity(actor_projection(women_group), type = "local"),
  triad_closure_opsahl(women_group, type = "local"),
  triad_closure_exclusive(women_group, type = "local"),
  dynamic_triad_closure_projection(women_group, type = "local"),
  dynamic_triad_closure(women_group, type = "local")
)

Wedge censuses and closure indicators for dynamic affiliation networks

Description

Given a dynamic affiliation network and an actor node ID, identify all wedges for a specified measure centered at the node and indicate whether each is closed.

Usage

dynamic_wedges(
  graph,
  actor,
  alcove = 0,
  wedge = 0,
  maps = 0,
  congruence = 0,
  memory = Inf,
  wedge.gap = Inf,
  close.after = 0,
  close.before = Inf
)

Arguments

Details

The ⁠dynamic_wedges_*⁠ functions implement wedge censuses underlying the several measures of triad closure described below. Each function returns a transversal of wedges from the congruence classes of wedges centered at the index actor and indicators of whether each class is closed. The shell function dynamic_wedges determines a unique measure from several coded arguments (see below) and passes the input affiliation network to that measure.

Value

A two-element list consisting of (1) a 3- or 5-row integer matrix of (representatives of) all (congruence classes of) wedges in graph centered at actor, and (2) a logical vector indicating whether each wedge is closed.

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

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.

See Also

Other wedge functions: indequ_wedges()

Affiliation network wedges

Description

Each clustering coefficient can be defined as the proportion of "wedges" that are "closed", for suitable definitions of both terms. These functions count the "wedges", and among them the "closed" ones, centered at a given actor node in a given affiliation network.

Given an affiliation network and an actor node ID, identify all wedges for a specified measure centered at the node and indicate whether each is closed.

Usage

indequ_wedges(graph, Q)

indequ.wedges(graph, Q)

indstr_wedges(graph, Q)

indstr.wedges(graph, Q)

injact_wedges(graph, Q)

injact.wedges(graph, Q)

injequ_wedges(graph, Q)

injequ.wedges(graph, Q)

injstr_wedges(graph, Q)

injstr.wedges(graph, Q)

wedges(graph, actor, alcove = 0, wedge = 0, maps = 0, congruence = 0)

wedges_an(graph, actor, alcove = 0, wedge = 0, maps = 0, congruence = 0)

wedges_watts_strogatz(graph, actor)

wedges_classical(graph, actor)

wedges_projection(graph, actor)

wedges_opsahl(graph, actor)

wedges_twomode(graph, actor)

wedges_liebig_rao_0(graph, actor)

wedges_unconnected(graph, actor)

wedges_liebig_rao_3(graph, actor)

wedges_completely_connected(graph, actor)

wedges_exclusive(graph, actor)

centered_triads(graph, actor)

triad_wedges_watts_strogatz(w, x, y, z)

triad_wedges_classical(w, x, y, z)

triad_wedges_projection(w, x, y, z)

triad_wedges_homact(w, x, y, z)

triad_wedges_opsahl(w, x, y, z)

triad_wedges_twomode(w, x, y, z)

triad_wedges_injequ(w, x, y, z)

triad_wedges_liebig_rao_0(w, x, y, z)

triad_wedges_unconnected(w, x, y, z)

triad_wedges_indequ(w, x, y, z)

triad_wedges_liebig_rao_1(w, x, y, z)

triad_wedges_sparsely_connected(w, x, y, z)

triad_wedges_liebig_rao_2(w, x, y, z)

triad_wedges_highly_connected(w, x, y, z)

triad_wedges_liebig_rao_3(w, x, y, z)

triad_wedges_completely_connected(w, x, y, z)

triad_wedges_exclusive(w, x, y, z)

triad_wedges_indstr(w, x, y, z)

triad_wedges_indact(w, x, y, z)

triad_wedges_homequ(w, x, y, z)

triad_wedges_homstr(w, x, y, z)

triad_wedges_injstr(w, x, y, z)

triad_wedges_injact(w, x, y, z)

triad_wedges(w, x, y, z, alcove = 0, wedge = 0, maps = 0, congruence = 0)

Arguments

Details

The ⁠wedges_*⁠ functions implement wedge censuses underlying the several measures of triad closure described below. Each function returns a transversal of wedges from the congruence classes of wedges centered at the index actor and indicators of whether each class is closed. The shell function wedges determines a unique measure from several coded arguments (see below) and passes the input affiliation network to that measure.

Value

A two-element list consisting of (1) a 3- or 5-row integer matrix of (representatives of) all (congruence classes of) wedges in graph centered at actor, and (2) a logical vector indicating whether each wedge is closed.

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

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.

See Also

Other wedge functions: dynamic_wedges()

Combinatorial bijections for affiliation network triad indexing

Description

These functions biject among partitions of at most 3 parts, 3-subsets of natural numbers, and indices for the lexicographic total orders on them.

Usage

index_subset(i)

subset_index(vec)

subset_partition(vec)

partition_subset(lambda)

index_partition(i)

partition_index(lambda)

indexSubset(i)

indexPartition(i)

subsetIndex(vec)

subsetPartition(vec)

partitionIndex(lambda)

partitionSubset(lambda)

Arguments

Value

Numeric vectors.

Examples

index_subset(2)
index_partition(2)
subset_index(c(3, 2, 0))
subset_partition(c(3, 2, 0))
partition_index(c(1, 1, 0))
partition_subset(c(1, 1, 0))

Interlocks among corporate philanthropists in Minneapolis-St. Paul.

Description

These data record the memberships of 26 directors on the boards of 15 companies. They are a subset of a larger dataset that is not publicly available.

Format

An affiliation network; see is_an.

Source

Wasserman, S. & Faust, K. (1994). Social Network Analysis: Methods and Applications. Cambridge University Press. http://books.google.com/books?id=CAm2DpIqRUIC

Add actor and event nodes

Description

These functions add actor and event nodes (as desired) to a graph while maintaining its (temporal) affiliation network structure.

Usage

add_modes(graph, mode = 1, nv, ..., attr = list(), affiliations = NULL)

add_actors(graph, nv, ..., attr = list(), events = NULL)

add_events(graph, nv, ..., attr = list(), actors = NULL)

Arguments

Value

The input graph with additional actor or event nodes added.

See Also

Original igraph functions: add_vertices(), add_edges()

Other modal queries and manipulations: dualize(), mode_counts, modes, schedule()

Examples

data(women_clique)
plot(prettify_an(add_actors(women_clique, nv = 1, events = c(7, 9))))
data(women_group)
plot(prettify_an(women_group))
actor_names <- c("Frances", "Dorothy")
cbind(
  dynamic_triad_closure(women_group, type = "local"),
  dynamic_triad_closure(
    add_events(women_group, nv = 1, actors = actor_names, time = 0),
    type = "local"
  ),
  dynamic_triad_closure(
    add_events(women_group, nv = 1, actors = actor_names, time = 367),
    type = "local"
  )
)

Count the actors and events in an affiliation network

Description

These functions return the number of actors (nodes with type attribute FALSE) or events (TRUE) in an affiliation network.

Usage

actor_count(graph)

event_count(graph)

actor.count(graph)

event.count(graph)

Arguments

Value

An integer.

See Also

Original igraph functions: vcount(), ecount()

Other modal queries and manipulations: dualize(), mode_addition, modes, schedule()

Examples

data(chicago1960s)
actor_count(chicago1960s)
event_count(chicago1960s)

Project an affiliation network onto its actors

Description

These functions use bipartite_projection() to compute the projections of an affiliation network onto the actor or event nodes.

Usage

mode_projection(graph, mode = 1, name = "name")

actor_projection(graph, ...)

event_projection(graph, ...)

actor.projection(graph, ...)

event.projection(graph, ...)

Arguments

Value

An igraph object with nodes corresponding to one "type" of the input graph.

See Also

Original igraph functions: bipartite_projection()

Examples

data(chicago1960s)
( tab <- table(V(chicago1960s)$type) )
( proj <- actor_projection(chicago1960s) )
vcount(proj) == tab[1]
( proj <- event_projection(chicago1960s) )
vcount(proj) == tab[2]

Actor and event node iteration and attribute querying and assignment

Description

These functions return actor and event node lists.

Usage

V1(graph)

V2(graph)

actor_attr(graph, name, index = V1(graph))

event_attr(graph, name, index = V2(graph))

set_actor_attr(graph, name, index = V1(graph), value)

set_event_attr(graph, name, index = V2(graph), value)

V1(x) <- value

V2(x) <- value

Arguments

Value

The value of the actor or event attribute, or the input graph with the attribute set.

See Also

Original igraph functions: V(), vertex_attr(), set_vertex_attr()

Other modal queries and manipulations: dualize(), mode_addition, mode_counts, schedule()

Examples

data(women_clique)
print(V1(women_clique))
print(V2(women_clique))
V1(women_clique)$label <- LETTERS[1:5]
V2(women_clique)$label <- 1:5
plot(prettify_an(women_clique))

Noordin Top meeting attendance network.

Description

These data record the attendance of 26 individuals at 20 meetings associated with Noordin Mohammad Top.

Format

An affiliation network; see is_an.

Source

Noordin Top Terrorist Network Data. (2011). The Association of Religious Data Archives. https://web.archive.org/web/20140625114607/http://www.thearda.com/Archive/Files/Descriptions/TERRNET.asp

Noordin Top organization membership network.

Description

These data record the attendance of 67 members of 32 organizations associated with Noordin Mohammad Top.

Format

An affiliation network; see is_an.

Source

Noordin Top Terrorist Network Data. (2011). The Association of Religious Data Archives. https://web.archive.org/web/20140625114607/http://www.thearda.com/Archive/Files/Descriptions/TERRNET.asp

Convenient plotting aesthetics for affiliation networks

Description

Given an affiliation network, assign the node and link aesthetics to values that produce a neater visualization through plot.igraph than the igraph defaults.

Usage

prettify_an(graph)

prettify.an(graph)

Arguments

Value

The input graph with aesthetic node and link attributes (re)set.

Examples

library(igraph)
data(women_clique)
data(whigs)
for (g in list(women_clique, whigs)) {
    plot(prettify_an(g))
}

Project a higher-resolution triad census to a lower-resolution one

Description

Given a triad census of any scheme, construct a triad census of a coarser (strictly less informative) scheme.

Usage

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)

Arguments

Details

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.

Value

A list of triad_census() outputs.

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.

See Also

Other triad census functions: triad_census(), triad_closure_from_census(), triad_tallies

Examples

data(women_group)
group_full_census <- triad_census(women_group, scheme = "full")
project_census(group_full_census, scheme = "full")
group_binary_census <- triad_census(women_group, scheme = "binary")
project_census(group_binary_census, scheme = "binary")

Affiliation network clustering coefficients

Description

Each clustering coefficient can be defined as the proportion of "wedges" that are "closed", for suitable definitions of both terms. The main function, transitivity_an(), calls one of the wedge functions and computes the global or local clustering coefficient of the given affiliation network, and if the local, then at the given nodes. (project_transitivity cheats by using transitivity() but produces output consistent with the other variants of transitivity_an.)

Usage

project_transitivity(graph, type = "global", vids = which(!V(graph)$type))

project.transitivity(graph, type = "global", vids = which(!V(graph)$type))

Arguments

Value

If type is "global", the global clustering coefficient of the network (a single numeric value); if "local", the local clustering coefficients of the actors (a numeric vector); otherwise, a 2-column matrix, each row of which gives the number of wedges and the number of closed wedges centered at each actor.

See Also

Other triad closure functions: dynamic_triad_closure(), transitivity_an(), triad_closure(), triad_closure_from_census()

Actors and their shared events

Description

Given an affiliation network and a vector of actor node IDs, produce the induced subgraph on the actor nodes together with all event nodes incident to at least two of them. This is called the actors' schedule.

Usage

schedule(graph, actors = V(graph)[V(graph)$type == FALSE])

Arguments

Value

An igraph object induced from the input graph.

See Also

Other modal queries and manipulations: dualize(), mode_addition, mode_counts, modes

Examples

data(women_clique)
schedule(women_clique, actors = V1(women_clique)[seq(3)])

Networks of interlocking directorates

Description

Five affiliation networks constructed from the multiple directors lists of the top >100 Scottish companies over five two-year intervals between 1904 and 1974.

Format

An affiliation network; see is_an.

Source

Scott, J., & Hughes, M. (1980). The Anatomy of Scottish Capital. Croom Helm. http://books.google.com/books?id=59mvAwAAQBAJ

Affiliation network clustering coefficients

Description

This function computes a given flavor of transitivity (triadic closure) on a given affiliation network. The calculations are performed locally. Each flavor is defined as a proportion of "wedges" that are "closed", for suitable definitions of both terms. The function transitivity_an is a shell that proceeds across actors and computes wedges using the provided wedgeFun. These functions count the "wedges", and among them the "closed" ones, centered at a given actor node in a given affiliation network. The triads method transitivity_an_triads first classifies every triad centered at each node. The appropriate formula then counts the wedges and closed wedges at each. The method is slower for a single flavor but can be used to produce multiple flavors with negligible additional computational cost. The wedges method transitivity_an_wedges relies on a separate "wedge function" for each statistic. The algorithm calls the appropriate wedge function to run over the necessary wedge centers and return a wedge count matrix, which is returned back into transitivity_an for outputting.

Usage

transitivity_an(
  graph,
  type = "global",
  wedgeFun,
  flavor,
  vids = which(!V(graph)$type),
  add.names = FALSE
)

transitivity_an_triads(graph, vids = which(!V(graph)$type), flavor)

transitivity_an_wedges(graph, vids = which(!V(graph)$type), wedgeFun)

transitivity.an(
  graph,
  type = "global",
  wedgeFun,
  flavor,
  vids = which(!V(graph)$type),
  add.names = FALSE
)

transitivity.an.triads(graph, vids = which(!V(graph)$type), flavor)

transitivity.an.wedges(graph, vids = which(!V(graph)$type), wedgeFun)

indequ_transitivity(graph, type = "global", vids = which(!V(graph)$type))

indequ.transitivity(graph, type = "global", vids = which(!V(graph)$type))

indstr_transitivity(graph, type = "global", vids = which(!V(graph)$type))

indstr.transitivity(graph, type = "global", vids = which(!V(graph)$type))

injact_transitivity(graph, type = "global", vids = which(!V(graph)$type))

injact.transitivity(graph, type = "global", vids = which(!V(graph)$type))

injequ_transitivity(graph, type = "global", vids = which(!V(graph)$type))

injequ.transitivity(graph, type = "global", vids = which(!V(graph)$type))

injstr_transitivity(graph, type = "global", vids = which(!V(graph)$type))

injstr.transitivity(graph, type = "global", vids = which(!V(graph)$type))

opsahl_transitivity(graph, type = "global", vids = which(!V(graph)$type))

opsahl.transitivity(graph, type = "global", vids = which(!V(graph)$type))

excl_transitivity(graph, type = "global", vids = which(!V(graph)$type))

excl.transitivity(graph, type = "global", vids = which(!V(graph)$type))

Arguments

Value

If type is "global", the global clustering coefficient of the network (a single numeric value); if "local", the local clustering coefficients of the actors (a numeric vector); otherwise, a 2-column matrix, each row of which gives the number of wedges and the number of closed wedges centered at each actor.

See Also

Other triad closure functions: dynamic_triad_closure(), project_transitivity(), triad_closure(), triad_closure_from_census()

Affiliation network triads

Description

These functions create and operate on triads in affiliation networks. In this context, a triad is the schedule of a subset of three distinct actors.

Usage

make_triad(
  lambda,
  w,
  actor_names = c("p", "q", "r"),
  event_names = if (sum(c(lambda, w)) == 0) c() else as.character(1:sum(c(lambda, w)))
)

is_triad(graph)

triad_class(
  graph,
  actors = V(graph)[V(graph)$type == FALSE],
  as.partition = TRUE,
  format = "list"
)

layout_triad(
  triad = NULL,
  lambda = NULL,
  w = NULL,
  scale = 0.3,
  angdir = -1,
  rot = -pi/2,
  rot_lambda = c(0, 0, 0),
  rot_w = pi/12
)

plot_triad(
  triad = NULL,
  lambda = NULL,
  w = NULL,
  layout = NULL,
  prettify = TRUE,
  cex = 1,
  scale = 0.3,
  angdir = -1,
  rot = -pi/2,
  rot_lambda = c(0, 0, 0),
  rot_w = pi/12,
  actor_names = c("p", "q", "r"),
  event_names = if (sum(c(lambda, w)) == 0) c() else as.character(1:sum(c(lambda, w))),
  xlim = NULL,
  ylim = NULL,
  ...
)

an_triad(...)

is.triad(graph)

triad.class(
  graph,
  actors = V(graph)[V(graph)$type == FALSE],
  as.partition = TRUE,
  format = "list"
)

an.triad(...)

layout.triad(
  triad = NULL,
  lambda = NULL,
  w = NULL,
  scale = 0.3,
  angdir = -1,
  rot = -pi/2,
  rot_lambda = c(0, 0, 0),
  rot_w = pi/12
)

plotTriad(
  triad = NULL,
  lambda = NULL,
  w = NULL,
  layout = NULL,
  prettify = TRUE,
  cex = 1,
  scale = 0.3,
  angdir = -1,
  rot = -pi/2,
  rot_lambda = c(0, 0, 0),
  rot_w = pi/12,
  actor_names = c("p", "q", "r"),
  event_names = if (sum(c(lambda, w)) == 0) c() else as.character(1:sum(c(lambda, w))),
  xlim = NULL,
  ylim = NULL,
  ...
)

Arguments

Value

An igraph object, a logical value, a matrix of plotting coordinates, or a list of summary parameters lambda and w.

Examples

tr <- make_triad(lambda = c(3,1,1), w = 2)
is_triad(tr)
triad_class(tr)
layout_triad(tr)
plot_triad(tr)

Triad census for affiliation networks

Description

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

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.

Value

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

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.

See Also

Original igraph functions: triad_census()

Other triad census functions: project_census(), triad_closure_from_census(), triad_tallies

Examples

data(women_clique)
(tc <- triad_census(women_clique, add.names = TRUE))
sum(tc) == choose(vcount(actor_projection(women_clique)), 3)

Triad closure for affiliation networks

Description

Given an affiliation network and a vector of actor node IDs, calculate a specified measure of triad closure centered at the nodes.

Usage

triad_closure(graph, ...)

triad_closure_an(graph, method = "wedges", ...)

triad_closure_via_triads(
  graph,
  actors = V(graph)[V(graph)$type == FALSE],
  type = "global",
  ...
)

triad_closure_from_centered_triads(
  triad_list,
  type = "global",
  ...,
  measure = NULL,
  triads.fun = NULL
)

triad_closure_via_wedges(
  graph,
  actors = V(graph)[V(graph)$type == FALSE],
  type = "global",
  ...,
  measure = NULL,
  wedges.fun = NULL
)

triad_closure_watts_strogatz(
  graph,
  actors = V(graph)[V(graph)$type == FALSE],
  type = "global"
)

triad_closure_classical(
  graph,
  actors = V(graph)[V(graph)$type == FALSE],
  type = "global"
)

triad_closure_opsahl(
  graph,
  actors = V(graph)[V(graph)$type == FALSE],
  type = "global"
)

triad_closure_twomode(
  graph,
  actors = V(graph)[V(graph)$type == FALSE],
  type = "global"
)

triad_closure_liebig_rao_0(
  graph,
  actors = V(graph)[V(graph)$type == FALSE],
  type = "global"
)

triad_closure_unconnected(
  graph,
  actors = V(graph)[V(graph)$type == FALSE],
  type = "global"
)

triad_closure_liebig_rao_3(
  graph,
  actors = V(graph)[V(graph)$type == FALSE],
  type = "global"
)

triad_closure_completely_connected(
  graph,
  actors = V(graph)[V(graph)$type == FALSE],
  type = "global"
)

triad_closure_exclusive(
  graph,
  actors = V(graph)[V(graph)$type == FALSE],
  type = "global"
)

triad_closure_projection(
  graph,
  actors = V(graph)[V(graph)$type == FALSE],
  type = "global"
)

Arguments

Details

The ⁠triad_closure_*⁠ functions implement the several measures of triad closure described below. Each function returns a single global statistic, a vector of local statistics, or a matrix of local denominators and numerators from which the global and local statistics can be recovered.

The function triad_closure_projection recapitulates triad_closure_watts_strogatz() by invoking the bipartite_projection() and transitivity() functions in igraph.

Value

If type is "global", the global statistic for graph (a single numeric value); if "local", the local statistics for actors (a numeric vector); if "raw", a 2-column matrix, each row of which gives the number of wedges and of closed wedges centered at actors.

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

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.

See Also

Original igraph functions: transitivity()

Other triad closure functions: dynamic_triad_closure(), project_transitivity(), transitivity_an(), triad_closure_from_census()

Examples

data(women_clique)
mapply(
  triad_closure,
  measure = c("classical", "twomode", "unconnected", "exclusive"),
  MoreArgs = list(graph = women_clique, type = "local")
)
data(women_group)
cbind(
  triad_closure_watts_strogatz(women_group, type = "local"),
  triad_closure_opsahl(women_group, type = "local"),
  triad_closure_liebig_rao_0(women_group, type = "local"),
  triad_closure_exclusive(women_group, type = "local")
)

Global triad closure from a triad census

Description

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

Usage

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

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.

Value

Output equivalent to that of triad_closure().

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.

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

See Also

Other triad census functions: project_census(), triad_census(), triad_tallies

Other triad closure functions: dynamic_triad_closure(), project_transitivity(), transitivity_an(), triad_closure()

Triad tallies

Description

These functions are called by the full triad census to handle triads of different types using the projection onto actor nodes. The name of each function indicates the number of edges that appear among the three actors of the triad in the projection. (Zero-edge triads do not need to be tallied; the total number of triads is easily calculated, and the difference between this number and the total number of triads with edges gives the number of triads without.)

Usage

connectedTriples(bigraph, graph = actor_projection(bigraph, name = "id"))

oneTiedTriads(graph)

twoTiedTriads(graph)

threeTiedTriads(bigraph, graph = actor_projection(bigraph, name = "id"))

Arguments

See Also

Other triad census functions: project_census(), triad_census(), triad_closure_from_census()

Membership network of American Whigs.

Description

These data record the membership of 136 colonial Americans in 5 Whig organizations.

Format

An affiliation network; see is_an.

Source

Fischer, D.H. (1994). Paul Revere's Ride. Oxford University Press. https://books.google.com/books?id=knC-kTFI9_gC https://github.com/kjhealy/revere

Clique of women connected by event coattendance in Old City.

Description

These data record the participation of 5 African-American women in 5 social activities.

Format

An affiliation network; see is_an.

Source

Davis, A., Gardner, B.B., & Gardner, M.R. (1941). Deep South: A Social Anthropological Study of Caste and Class. Columbia, SC: University of South Carolina Press. http://books.google.com/books?id=HGIdAAAAIAAJ

Group of women connected by event coattendance in Old City.

Description

These data record the attendance of 18 white women at 14 social events over a 9-month period.

Format

An affiliation network; see is_an. Events carry a time attribute equal to the number of days since the previous 31 December.

Source

Davis, A., Gardner, B.B., & Gardner, M.R. (1941). Deep South: A Social Anthropological Study of Caste and Class. Columbia, SC: University of South Carolina Press. http://books.google.com/books?id=HGIdAAAAIAAJ