Data
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
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:
- Group membership of the ego
- Group membership of the alter
- Whether or not ego and alter are directly connected
Formally, mixing matrix is a matrix \(M\) in which entry \(m_{ghy}\) is a number of pairs of nodes
such that
- The first node belongs to group \(g\)
- The second node belongs to group \(h\)
- \(y\) is
TRUE if there
is a tie, \(y\) is FALSE
if there is no tie
We 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:
mixingm(Classroom, "gender")
#> alter
#> ego Boy Girl
#> Boy 40 2
#> Girl 5 41
Among other things we see that:
- There are \(40 + 41 = 81\) ties
within groups.
- There are only \(5 + 2 = 7\) ties
between groups.
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:
- What is the probability of a tie depending on attributes of
nodes?
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
- What is the distribution of group memberships of alters depending on
the attribute of ego?
round( prop.table(m[,,2], 1 ) * 100, 1)
#> alter
#> ego Boy Girl
#> Boy 95.2 4.8
#> Girl 10.9 89.1
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.
Mixing data
frames
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:
mixingdf(undir, "gender")
#> alter ego n
#> 1 Boy Boy 11
#> 3 Boy Girl 1
#> 4 Girl Girl 10
mixingdf(undir, "gender", full=TRUE)
#> alter ego tie n
#> 1 Boy Boy FALSE 67
#> 3 Boy Girl FALSE 168
#> 4 Girl Girl FALSE 68
#> 5 Boy Boy TRUE 11
#> 7 Boy Girl TRUE 1
#> 8 Girl Girl TRUE 10
