Wedge censuses and closure indicators for dynamic affiliation networks

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

graph: A dynamic affiliation network.
actor: An actor node in graph.
alcove, wedge, maps, congruence: Choice of alcove, wedge, maps, and congruence (see Details).
memory: Numeric; minimum delay of wedge formation since would-have-been closing events.
wedge.gap: Numeric; maximum delay between the two events of a wedge.
close.after, close.before: Numeric; minimum and maximum delays after both events form a wedge for a third event to close it.

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.

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.

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.