The following vignette demonstrates using the functions from package netseg (Bojanowski 2021). Two example datasets are described in the next section. Mixing matrices are described in section 2 and the measures are described in section 3. Please consult Bojanowski and Corten (2014) for further details.
In the examples below we will use data Classroom
, a
directed network in a classroom of 26 kids (Dolata, n.d.). Ties correspond to nominations
from a survey question “With whom do you like to play with?”. Here is a
picture:
plot(
Classroom,
vertex.color = c("Skyblue", "Pink")[match(V(Classroom)$gender, c("Boy", "Girl"))],
vertex.label = NA,
vertex.size = 10,
edge.arrow.size = .7
)
legend(
"topright",
pch = 21,
legend = c("Boy", "Girl"),
pt.bg = c("Skyblue", "Pink"),
pt.cex = 2,
bty = "n"
)
For us it will be a graph \(G = <V, E>\) where the node-set \(V = \{1, ..., i, ..., N\}\) correspond to kids, and edges \(E\) correspond to “play-with” nominations. Additionally, we need a node attribute, say \(X\), exhaustivelty assigning nodes to mutually-exclusive \(K\) groups. In the classroom example \(X\) is gender with values “Boy” and “Girl” (so \(K=2\)).
Some measures are applicable only to an undirected network. For that
purpose let’s create an undirected network of reciprocated nominations
in the Classroom
network and call it
undir
:
undir <- as.undirected(Classroom, mode="mutual")
plot(
undir,
vertex.color = c("Skyblue", "Pink")[match(V(undir)$gender, c("Boy", "Girl"))],
vertex.label = NA,
vertex.size = 10,
edge.arrow.size = .7
)
legend(
"topright",
pch = 21,
legend = c("Boy", "Girl"),
pt.bg = c("Skyblue", "Pink"),
pt.cex = 2,
bty = "n"
)
Mixing matrix is traditionally a two-dimensional cross-classification of edges depending on group membership of the adjacent nodes. A three-dimensional version of a mixing matrix cross-classifies all the dyads according to the following criteria:
Formally, mixing matrix is a matrix \(M\) in which entry \(m_{ghy}\) is a number of pairs of nodes such that
TRUE
if there
is a tie, \(y\) is FALSE
if there is no tieWe can compute the mixing matrix for the classroom network and
attribute gender
with the function mixingm()
.
By default the traditional two-dimensional version is returned:
Among other things we see that:
Supplying argument full=TRUE
the function will return an
three-dimensional array cross-classifying the dyads:
m <- mixingm(Classroom, "gender", full=TRUE)
m
#> , , tie = FALSE
#>
#> alter
#> ego Boy Girl
#> Boy 116 167
#> Girl 164 115
#>
#> , , tie = TRUE
#>
#> alter
#> ego Boy Girl
#> Boy 40 2
#> Girl 5 41
We can analyze the mixing matrix as a typical frequency crosstabulation. For example:
round( prop.table(m, c(1,2)) * 100, 1)
#> , , tie = FALSE
#>
#> alter
#> ego Boy Girl
#> Boy 74.4 98.8
#> Girl 97.0 73.7
#>
#> , , tie = TRUE
#>
#> alter
#> ego Boy Girl
#> Boy 25.6 1.2
#> Girl 3.0 26.3
In other words, boys are 95% of nominations of other boys, but only 11% of nominations of girls.
Function mixingm()
works also for undirected networks,
values below the diagonal are always 0:
mixingm(undir, "gender")
#> ego
#> alter Boy Girl
#> Boy 11 1
#> Girl 0 10
mixingm(undir, "gender", full=TRUE)
#> , , tie = FALSE
#>
#> ego
#> alter Boy Girl
#> Boy 67 168
#> Girl 0 68
#>
#> , , tie = TRUE
#>
#> ego
#> alter Boy Girl
#> Boy 11 1
#> Girl 0 10
Most of the segregation indexes described below summarize the mixing matrix.
Function mixingdf()
returns the same data in the form of
a data frame. For directed Classroom
network:
mixingdf(Classroom, "gender")
#> ego alter n
#> 1 Boy Boy 40
#> 2 Girl Boy 5
#> 3 Boy Girl 2
#> 4 Girl Girl 41
mixingdf(Classroom, "gender", full=TRUE)
#> ego alter tie n
#> 1 Boy Boy FALSE 116
#> 2 Girl Boy FALSE 164
#> 3 Boy Girl FALSE 167
#> 4 Girl Girl FALSE 115
#> 5 Boy Boy TRUE 40
#> 6 Girl Boy TRUE 5
#> 7 Boy Girl TRUE 2
#> 8 Girl Girl TRUE 41
For undir
:
Coleman’s index compares the distribution of group memberships of alters with the distribution of group sizes. It captures the extent the nominations are “biased” due to the preference for own group.
Values are close to 1 (high segregation). The value for boys is greater than for girls, so girls nominated boys a bit more often than boys nominated girls.
Is applicable to undirected networks with two groups.
Function freeman
:
Values for vertices
(v <- ssi(undir, "gender"))
#> 1 2 3 4 5 6 7 8
#> 1.1392193 0.7033670 0.9816498 1.0000000 1.0000000 0.9973715 1.0000000 0.0000000
#> 9 10 11 12 13 14 15 16
#> 0.0000000 0.0000000 0.0000000 0.0000000 0.7151930 0.6925726 0.0000000 1.0333177
#> 17 18 19 20 21 22 23 24
#> 0.9701057 1.0344291 1.0550505 1.0299471 1.0000000 1.0000000 1.0000000 1.1031876
#> 25 26
#> 0.0000000 0.9816498
Plotted with grayscale (the more segregated the darker the color):
kol <- gray(scales::rescale(v, 1:0))
plot(
undir,
vertex.shape = c("circle", "square")[match(V(undir)$gender, c("Boy", "Girl"))],
vertex.color = kol,
vertex.label = V(undir),
vertex.label.color = ifelse(apply(col2rgb(kol), 2, mean) > 125, "black", "white"),
vertex.size = 15,
vertex.label.family = "sans",
edge.arrow.size = .7
)