Triad closure for affiliation networks

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

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

graph	An affiliation network.
...	Measure specifications passed to `wedges`.
method	Character; for a given `measure`, whether to use the measure-specific wedge census (`"wedges"`) or the measure-specific calculation on the centered triad census (`"triads"`).
actors	A vector of actor nodes in `graph`.
type	The type of statistic, matched to `"global"`, `"local"`, or `"raw"`.
triad_list	A list of triad isomorphism classes in matrix format, as produced by `centered_triads`.
measure	Character; the measure of triad closure, used as the suffix `` to `triad_closure_` Matched to `"classical"` (also `"watts_strogatz"`), `"twomode"` (also `"opsahl"`), `"unconnected"` (also `"liebig_rao_0"`), `"completely_connected"` (also `"liebig_rao_3"`), or `"exclusive"`.
triads.fun	A custom triad closure calculation. It must accept a vector of centered triad isomorphism classes, encoded as vectors `w`, `x`, `y`, and `z`, and return a 2-row integer matrix recording the number of wedges of the desired measure centered at the second actor, and involving the other two actors, of each triad.
wedges.fun	A custom wedge census function. It must accept an affiliation network `graph` and a single actor node ID `actor` and may have any additional parameters. It must return a named list with values `wedges` a numeric matrix of node IDs whose columns record the wedges centered at `actor` and `closed` a logical vector recording whether each wedge is closed. Overrides `measure`.

Value

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

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.

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.

Examples

data(women_clique)
mapply(
  triad_closure,
  measure = c("classical", "twomode", "unconnected", "exclusive"),
  MoreArgs = list(graph = women_clique, type = "local")
)
#>      classical   twomode unconnected exclusive
#> [1,] 0.8333333 0.5000000   0.5000000      0.50
#> [2,] 1.0000000 0.6666667   1.0000000      1.00
#> [3,] 1.0000000 0.6666667   0.6666667      0.50
#> [4,] 0.8333333 0.6000000   0.5000000      0.50
#> [5,] 0.8333333 0.7142857   0.6666667      0.75
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")
)
#>            [,1]      [,2]       [,3]      [,4]
#>  [1,] 0.8970588 0.7666667 0.39574468 0.4477612
#>  [2,] 0.9619048 0.8421751 0.44680851 0.4871795
#>  [3,] 0.8970588 0.7523437 0.06188925 0.1445783
#>  [4,] 0.9619048 0.8387909 0.34545455 0.4500000
#>  [5,] 1.0000000 1.0000000 1.00000000 1.0000000
#>  [6,] 0.9619048 0.8690476 0.66666667 0.7777778
#>  [7,] 0.9619048 0.7959184 0.50943396 0.5312500
#>  [8,] 0.9333333 0.6462585 0.40740741 0.4666667
#>  [9,] 0.8970588 0.6702509 0.28688525 0.3283582
#> [10,] 0.8970588 0.6740891 0.37777778 0.3928571
#> [11,] 0.9333333 0.7138810 0.67346939 0.5555556
#> [12,] 0.9333333 0.7695560 0.72602740 0.5357143
#> [13,] 0.8970588 0.7461929 0.33950617 0.3000000
#> [14,] 0.8970588 0.8379501 0.71854305 0.6631579
#> [15,] 0.8970588 0.8159204 0.71428571 0.6610169
#> [16,] 0.9333333 0.5407407 0.46666667 0.4666667
#> [17,] 1.0000000 0.5806452 1.00000000 1.0000000
#> [18,] 1.0000000 0.5806452 1.00000000 1.0000000