Fly.Rmd

---
title: 'Using R in Teaching from *Network Science*'
author: "Amir Barghi, Department of Mathematics and Statistics, Saint Michael's College"
output:
  html_document:
    df_print: paged
  word_document: default
  pdf_document: default
---

# Fly (*Drosophila Medulla*) Connectome

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

# Loading Packages

```{r, message = FALSE}
library(tidyverse)

library(igraph)

library(igraphdata)

library(ggraph)

library(latex2exp)
```

# Loading the Data Set

## Data from [NeuroData's Graph DataBase](https://neurodata.io/project/connectomes/)

Data Source: Takemura, Sy., Bharioke, A., Lu, Z. et al. A visual motion detection circuit suggested by Drosophila connectomics. *Nature* **500**, 175–181 (2013). https://doi.org/10.1038/nature12450

```{r}
g <- read_graph('https://s3.amazonaws.com/connectome-graphs/fly/drosophila_medulla_1.graphml', 
                format = 'graphml')
```

```{r}
V(g)

E(g)

components(g)$no

components(g)$csize

glimpse(vertex_attr(g))

glimpse(edge_attr(g))

vertex_attr(g, name = 'name')[1:10]

edge_attr(g, name = 'proofreading.details')[1:10]
```

# Visualizing the Fly Connectome

```{r}
set.seed(42)

ggraph(g, layout = 'lgl') + 
  geom_edge_fan(edge_linetype = 3, color = 'dark blue', alpha = 0.25) + 
  geom_node_point(color = 'dark red', size = 1, alpha = 0.75) + 
  theme_graph(base_family = 'Helvetica') +
  labs(title = 'Fly Connectome',
       subtitle = 'Displayed Using Layout Generator for Larger Graphs')
```

```{r}
set.seed(42)

ggraph(g, layout = 'drl') + 
  geom_edge_fan(edge_linetype = 3, color = 'dark blue', alpha = 0.25) + 
  geom_node_point(color = 'dark red', size = 1, alpha = 0.75) + 
  theme_graph(base_family = 'Helvetica') +
  labs(title = 'Fly Connectome',
       subtitle = 'Displayed Using Distributed Recursive Layout')
```

```{r}
set.seed(42)

ggraph(g, layout = 'mds') + 
  geom_edge_fan(edge_linetype = 3, color = 'dark blue', alpha = 0.25) + 
  geom_node_point(color = 'dark red', size = 1, alpha = 0.75) + 
  theme_graph(base_family = 'Helvetica') +
  labs(title = 'Fly Connectome',
       subtitle = 'Displayed Using Multidimensional Scaling Layout')
```

# Summary Statistics of the Fly Connectome

```{r}
suppressMessages(df <- bind_cols(enframe(eccentricity(g, mode = 'out')), 
                                 enframe(betweenness(g)), 
                                 enframe(degree(g, mode = 'out')), 
                                 enframe(transitivity(g, type = c('local')))))

df <- df %>% select(name...1, value...2, value...4, value...6, value...8)

names(df) <- c('name', 'eccentricity', 'betweenness', 'outdegree', 'clustering')

head(df)

tail(df)

glimpse(df)
```

```{r}
df %>% 
  summarize(avg_deg = mean(outdegree), 
            delta = max(outdegree), 
            prop = sum(outdegree <= avg_deg) / n(), 
            diam = max(eccentricity),
            radius = min(eccentricity),
            avg_cc = mean(clustering, na.rm = TRUE),
            avg_distance = mean_distance(g, directed = TRUE, unconnected = TRUE))
```

```{r}
(d <- mean_distance(g, directed = TRUE, unconnected = TRUE))

mean(distances(g))
```

## Fig. 2.18(a) on p. 66

```{r}
distance_table(g)

D <- data.frame(1:length(distance_table(g)$res), 
                distance_table(g)$res / sum(distance_table(g)$res))

names(D) <- c('x', 'y')

D %>% 
  ggplot(aes(x = x, y = y)) + 
  geom_point() + 
  geom_line(aes(x = d), color = 'blue') +
  labs(title = 'Distribution of Distance (Proportions) in the Fly Connectome') +
  labs(x = 'distance', y = 'density')
```

## The Outdegree Distribution 

```{r}
df %>% 
  ggplot(aes(x = outdegree, y = ..density..)) + 
  geom_density(fill = 'red') +
  labs(title = 'KDE of Outdegrees in the Fly Connectome')
```

```{r}
df %>% 
  ggplot(aes(x = outdegree, y = ..density..)) + 
  geom_histogram(binwidth = 1, fill = 'blue') +
  labs(title = 'Histogram of Outdegrees in the Fly Connectome')
```

```{r}
df %>% 
  filter(outdegree <= 20) %>% 
  ggplot(aes(x = outdegree, y = ..density..)) + 
  geom_density(fill = 'red') + 
  labs(title = 'KDE of Outdegrees in the Fly Connectome',
       subtitle = TeX('for Nodes with Outdegree $\\leq 20$'))
```

```{r}
df %>% 
  filter(outdegree <= 20) %>% 
  ggplot(aes(x = outdegree, y = ..density..)) + 
  geom_histogram(binwidth = 1, fill = 'blue') +
  labs(title = 'Histogram of Outdegrees in the Fly Connectome',
       subtitle = TeX('for Nodes with Outdegree $\\leq 20$'))
```

## Fig. 2.18(b) on p. 66

```{r, message = FALSE}
df %>% 
  group_by(outdegree) %>% 
  summarise(cc_deg = mean(clustering, na.rm = TRUE)) %>%
  ungroup() %>%
  filter(outdegree > 0) %>%
  ggplot(aes(x = outdegree, y = cc_deg)) + 
  geom_point(na.rm = TRUE, color = 'blue') + 
  scale_x_log10() + 
  scale_y_log10() +
  labs(title = 'Relation Between Local Clustering Coefficient and Outdegree',
       subtitle = 'in the Fly Connectome') +
  labs(x = TeX('$p_k$'), y = TeX('$C_k$'))
```

## Local Clustering Coefficient Distribution

```{r}
df %>% 
  ggplot(aes(x = clustering, y = ..density..)) + 
  geom_density(fill = 'red', na.rm = TRUE) +
  labs(title = 'KDE of Local Clustering Coefficients in the Fly Connectome')
```  

```{r}
df %>% 
ggplot(aes(x = clustering, y = ..density..)) + 
  geom_histogram(binwidth = .1, fill = 'blue', na.rm = TRUE) +
  labs(title = 'Histogram of Local Clustering Coefficients in the Fly Connectome')
```

```{r}
log(gorder(g)) / log(mean(df$outdegree))

mean_distance(g, directed = TRUE, unconnected = TRUE)

diameter(g)
```

```{r, message = FALSE}
C <- mean(df$clustering, na.rm = TRUE)

M <- mean(df$outdegree)

df %>%
  group_by(outdegree) %>% 
  summarise(cc_deg = mean(clustering)) %>%
  ungroup()
```

## Fig. 3.13(d) on p. 96

```{r, message = FALSE}
df %>% 
  group_by(outdegree) %>% 
  summarise(cc_deg = mean(clustering)) %>%
  filter(outdegree > 0) %>%
  ggplot(aes(x = outdegree, y = cc_deg)) + 
  geom_point(na.rm = TRUE, color = 'blue') + 
  geom_line(aes(y = C), color = 'blue') + 
  geom_line(aes(y = M / gorder(g)), color = 'red') + 
  scale_x_log10() + 
  scale_y_log10() + 
  labs(title =  'Relation Between Local Clustering Coefficient and Outdegree',
       subtitle = 'The blue line is the average local clustering coefficient; \nthe red one is the one predicted by the random model.') +
  labs(x = 'k', y = TeX('$C(k)$'))
```

## Visualizing Other Relations with Outdegree

```{r}
df %>%
  filter(outdegree > 0) %>%
  ggplot(aes(x = outdegree, y = betweenness)) + 
  geom_point(na.rm = TRUE, size = 0.5, color = 'red') + 
  scale_x_log10() +
  labs(title = 'Relationship Between Betweenness Centrality and Outdegree') +
  labs(x = TeX('$\\log_{10}$(outdegree)'))
```

```{r}
df %>% 
  filter(outdegree > 0) %>%
  ggplot(aes(x = outdegree, y = betweenness + 0.00000001)) + 
  geom_point(na.rm = TRUE, size = 0.5, color = 'red') + 
  scale_y_log10() +
  labs(title = TeX('Relationship Between $\\log_{10}$ of Betweenness Centrality and Outdegree')) +
  labs(y = TeX('$\\log_{10}$(betweenness)'))
```

```{r}
df %>% 
  filter(betweenness > 0, outdegree > 0) %>% 
  ggplot(aes(x = outdegree, y = betweenness)) + 
  geom_point(na.rm = TRUE, size = 0.5, color = 'red') + 
  scale_y_log10() +
  scale_x_log10() +
  labs(title = TeX('Relationship Between $\\log_{10}$ of Betweenness Centrality and $\\log_{10}$ of Outdegree')) +
  labs(y = TeX('$\\log_{10}$(betweenness)'),
       x = TeX('$\\log_{10}$(outdegree)'))
```

```{r}
df %>% 
  filter(outdegree > 0) %>%
  ggplot(aes(x = outdegree, y = eccentricity)) + 
  geom_point(na.rm = TRUE, size = 0.5, color = 'orange') + 
  scale_x_log10() +
  labs(title = TeX('Relationship Between Eccentricity and $\\log_{10}$ of Outdegree')) +
  labs(x = TeX('$\\log_{10}$(outdegree)'))

```

```{r}
df %>% 
  filter(outdegree > 0) %>%
  ggplot(aes(x = outdegree, y = clustering)) + 
  geom_point(na.rm = TRUE, size = 0.5, color = 'blue') +
  scale_x_log10() +
  labs(title = TeX('Relationship Between Local Clustering Coefficient and $\\log_{10}$ of Outdegree')) +
  labs(x = TeX('$\\log_{10}$(outdegree)'))
```

# References

- Albert-Laszlo Barabasi, *Network Science*, Cambridge University Press, 2016. *Network Science* is available online at http://networksciencebook.com/ under the following license: "This book's text and illustrations are licensed under a [Creative Commons Attribution-NonCommercial 3.0 Unported License](https://creativecommons.org/licenses/by-nc/3.0/)."
- Takemura, Sy., Bharioke, A., Lu, Z. et al. A visual motion detection circuit suggested by Drosophila connectomics. *Nature* **500**, 175–181 (2013). https://doi.org/10.1038/nature12450