Network Segregation and Homophily

In netseg: Measures of Network Segregation and Homophily

library(netseg)
library(igraph)
requireNamespace("scales")

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  out.width = "100%",
  fig.width=10,
  fig.height=6
)

set.seed(666)

The following vignette demonstrates using the functions from package netseg [@r-netseg]. 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-corten-2014 for further details.

Data

data(Classroom)

In the examples below we will use data Classroom, a directed network in a classroom of r vcount(Classroom) kids [@dolata2014]. 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 = $ 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:

1. Group membership of the ego
2. Group membership of the alter
3. 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")

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

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)

• What is the distribution of group memberships of alters depending on the attribute of ego?

round( prop.table(m[,,2], 1 ) * 100, 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")
mixingm(undir, "gender", full=TRUE)

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")
mixingdf(Classroom, "gender", full=TRUE)

For undir:

mixingdf(undir, "gender")
mixingdf(undir, "gender", full=TRUE)

Measures

Assortativity coefficient

assort(Classroom, "gender")
assort(undir, "gender")

Coleman's homophily index

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.

• We have a separate value for each group
• Values are in [-1; 1]
• 0 -- Members of the given group nominate their group peers proportionally to the relative group size.
• 1 -- All nominations are from own group.
• -1 -- All nominations are from groups other than own.

coleman(Classroom, "gender")

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.

E-I

ei(Classroom, "gender")
ei(undir, "gender")

Freeman's segregation index

Is applicable to undirected networks with two groups.

• Values in [0;1]

Function freeman:

freeman(undir, "gender")

Gupta-Anderson-May

gamix(Classroom, "gender")
gamix(undir, "gender")

Odds-ratio

orwg(Classroom, "gender")
orwg(undir, "gender")

Segregation Matrix Index

smi(Classroom, "gender")

Spectral segregation index

Values for vertices

(v <- ssi(undir, "gender"))

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
)

References

netseg documentation built on July 9, 2023, 6:33 p.m.