---
title: "Network Segregation and Homophily"
author: "Michał Bojanowski"
date: "February 15, 2021"
output: 
  rmarkdown::html_vignette:
    toc: true
    number_sections: true
vignette: >
  %\VignetteIndexEntry{Network Segregation and Homophily}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
editor_options: 
  chunk_output_type: console
bibliography: refs.bib
---

```{r setup, include = FALSE}
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

```{r load-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:

```{r plot-data}
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`:

```{r data-undirected}
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:

```{r mixing-matrix-2d}
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:

```{r mixing-matrix-3d}
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?

```{r mm_condprob}
round( prop.table(m, c(1,2)) * 100, 1)
```

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

```{r mm_contact_layer}
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:

```{r mixingm-undir}
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:

```{r mixingdf-classroom}
mixingdf(Classroom, "gender")
mixingdf(Classroom, "gender", full=TRUE)
```

For `undir`:

```{r mixingdf-undir}
mixingdf(undir, "gender")
mixingdf(undir, "gender", full=TRUE)
```


# Measures

## Assortativity coefficient

```{r assort}
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.


```{r coleman}
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

```{r ei}
ei(Classroom, "gender")
ei(undir, "gender")
```


## Freeman's segregation index

Is applicable to undirected networks with two groups.

- Values in [0;1]



Function `freeman`:

```{r freeman}
freeman(undir, "gender")
```



## Gupta-Anderson-May

```{r gamix}
gamix(Classroom, "gender")
gamix(undir, "gender")
```

## Odds-ratio

```{r orwg}
orwg(Classroom, "gender")
orwg(undir, "gender")
```


## Segregation Matrix Index

```{r smi}
smi(Classroom, "gender")
```

## Spectral segregation index

Values for vertices

```{r ssi}
(v <- ssi(undir, "gender"))
```

Plotted with grayscale (the more segregated the darker the color):

```{r ssi-plot}
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