dynamic_wedges: Wedge censuses and closure indicators for dynamic affiliation...

In corybrunson/bitriad: Triadic Analysis of 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

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.

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

corybrunson/bitriad documentation built on May 13, 2019, 10:51 p.m.