Structural Equivalence
Description
Computes structural equivalence between ego and alter in a network
Usage
1 2 3 4 5 
Arguments
graph 
Any class of accepted graph format (see 
v 
Numeric scalar. Cohesion constant (see details). 
inf.replace 
Numeric scalar scalar. Replacing inf values obtained from 
groupvar 
Either a character scalar (if 
mode 
Character scalar pased to 
... 
Further arguments to be passed to 
x 
A 
Details
Structure equivalence is computed as presented in Valente (1995), and Burt (1987), in particular
SE(ij) = [dmax(i)  d(ji)]^v/[∑_k (dmax(i)  d(ki))^v]
with the summation over k!=i, and d(ji), Eucledian distance in terms of geodesics, is defined as
d(ji) = [(z(ji)  z(ij))^2 + ∑_k (z(jk)  z(ik))^2 + ∑_k (z_(ki)  z_(kj))^2]^(1/2)
with z(ij) as the geodesic (shortest path) from i to j, and dmax(i) equal to largest Euclidean distance between i and any other vertex in the network. All summations are made over k!={i,j}
Here, the value of v is interpreted as cohesion level. The higher its value, the higher will be the influence that the closests alters will have over ego (see Burt's paper in the reference).
Structural equivalence can be computed either for the entire graph or by groups
of vertices. When, for example, the user knows before hand that the vertices
are distributed accross separated communities, he can make this explicit to
the function and provide a groupvar
variable that accounts for this.
Hence, when groupvar
is not NULL
the algorithm will compute
structural equivalence within communities as marked by groupvar
.
Value
If graph
is a static graph, a list with the following elements:

Matrix of size n * n with Structural equivalence 

Matrix of size n * n Euclidean distances 

Matrix of size n * n Normalized geodesic distance 
In the case of dynamic graph, is a list of size t
in which each element
contains a list as described before. When groupvar
is specified, the
resulting matrices will be of class dgCMatrix
,
otherwise will be of class matrix
.
Author(s)
George G. Vega Yon, Stephanie R. Dyal, Timothy B, Hayes, Thomas W. Valente
References
Burt, R. S. (1987). "Social Contagion and Innovation: Cohesion versus Structural Equivalence". American Journal of Sociology, 92(6), 1287–1335. http://doi.org/10.1086/228667
Valente, T. W. (1995). "Network models of the diffusion of innovations" (2nd ed.). Cresskill N.J.: Hampton Press.
See Also
Other statistics: classify_adopters
,
cumulative_adopt_count
, dgr
,
ego_variance
, exposure
,
hazard_rate
, infection
,
moran
, threshold
,
vertex_covariate_dist
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23  # Computing structural equivalence for the fakedata 
data(fakesurvey)
# Coercing it into a diffnet object
fakediffnet < survey_to_diffnet(
fakesurvey, "id", c("net1", "net2", "net3"), "toa", "group"
)
# Computing structural equivalence without specifying group
se_all < struct_equiv(fakediffnet)
# Notice that pairs of individuals from different communities have
# nonzero values
se_all
se_all[[1]]$SE
# ... Now specifying a groupvar
se_group < struct_equiv(fakediffnet, groupvar="group")
# Notice that pairs of individuals from different communities have
# only zero values.
se_group
se_group[[1]]$SE
