Using R in Teaching from Network Science
Amir Barghi, Department of Mathematics and Statistics, Saint Michael’s College
Chap. 3: Random Networks – Erdos-Renyi
# configuring figure size
options(repr.plot.width = 6, repr.plot.height = 4)Loading Packages
library(tidyverse)
library(igraph)
library(ggraph)
library(glue)
library(latex2exp)3.2 The Random Network Model
\(G(N, \, p)\) Model
set.seed(42)
# the order of the graph
n <- 10
# the probability that a pair of nodes are adjacent
p <- 0.5
# creating an example of G(n, p) model
g <- sample_gnp(n, p)
# using igraph to visualize a graph
plot(g,
layout = layout_in_circle,
vertex.size = 2,
vertex.label = NA,
edge.lty = 3,
main = TeX(glue('An Example of $G(N, \\, p)$ Model for $N = $', n, ' and $p = $', p)))
# using ggraph to visualize a graph
ggraph(g, layout = 'circle') +
geom_edge_fan(edge_linetype = 3, color = 'dark blue', alpha = 0.25) +
geom_node_point(color = 'dark red', size = 2, alpha = 0.75) +
labs(title = TeX(glue('An Example of $G(N, \\, p)$ Model for $N = $', n, ' and $p = $', p))) +
theme_graph(base_family = 'Helvetica')
\(G(N, \, L)\) Model
set.seed(42)
# the order of the graph
n <- 10
# the probability that a pair of nodes are adjacent
p <- 0.5
# the size of the graph
m <- 22
# equation 3.0 on p. 75
choose(n, 2) * p## [1] 22.5# creating an example of G(n, l) model
g <- sample_gnm(n, m)
# using igraph to visualize g
plot(g,
layout = layout_in_circle,
vertex.size = 2,
vertex.label = NA,
edge.lty = 3,
main = TeX(glue('An Example of $G(N, \\, L)$ Model for $N = $', n, ' and $L = $', m)))
# using ggraph to visualize g
ggraph(g, layout = 'circle') +
geom_edge_link(edge_linetype = 3, color = 'dark blue', alpha = 0.25) +
geom_node_point(color = 'dark red', size = 2, alpha = 0.75) +
labs(title = TeX(glue('An Example of $G(N, \\, L)$ Model for $N = $', n, ' and $L = $', m))) +
theme_graph(base_family = 'Helvetica')
Note that \(\binom{n}{2} \cdot p = \binom{10}{2} \cdot 0.5 = 45 \cdot 0.5 \approx 22\).
3.4 Degree Distribution
3.4.1 Binomial Distribution
for (n in c(10, 20, 50)) {
for (p in c(0.1, 0.2, 0.3, 0.4, 0.5)) {
# possible values for X when X ~ Binom(n, p)
x <- c(0:n)
# P(X = x) when X ~ Binom(n, p)
y <- dbinom(x, n, p)
df <- data.frame(x, y)
names(df) <- c('x', 'y')
print(df %>%
ggplot(aes(x = x, y = y)) +
geom_point() +
geom_line(aes(x = n * p), color = 'blue') +
labs(title = TeX(glue('The Binomial Density Function for $n = $', n, ' and $p = $', p)),
subtitle = TeX(glue('$E\\[X\\] = n \\cdot p = $', n * p, ' (the blue vertical line)')),
y = TeX('P(X = x)')))
}
}
for (n in c(100, 500, 1000)) {
for (p in c(0.2, 0.5, 0.8)) {
x <- c(0:n)
y <- dbinom(x, n, p)
df <- data.frame(x, y)
names(df) <- c('x', 'y')
print(df %>%
ggplot(aes(x = x, y = y)) +
geom_bar(stat = 'identity', col = 'red', fill = 'red') +
geom_line(aes(x = n * p), color = 'blue') +
labs(title = TeX(glue('The Binomial Density Function for $n = $', n, ' and $p = $', p)),
subtitle = TeX(glue('$E\\[X\\] = n \\cdot p =$', n * p, ' (the blue vertical line)')),
y = TeX('P(X = x)')))
}
}
3.4.2 Poisson Distribution
for (n in c(10, 20, 50)) {
for (p in c(0.1, 0.2, 0.3, 0.4, 0.5)) {
x <- c(0:n)
# P(X = x) when X ~ Pois(n, p) and X <= n; lambda = n * p
y <- dpois(x, n * p)
df <- data.frame(x, y)
names(df) <- c('x', 'y')
print(df %>%
ggplot(aes(x = x, y = y)) +
geom_point() +
geom_line(aes(x = n * p), color = 'blue') +
labs(title = TeX(glue('The Poisson Density Function for $\\lambda$ = ', n * p,
' (n = ', n, ' and p = ', p, ')')),
subtitle = TeX(glue('$E\\[X\\] = \\lambda = $', n * p, ' (the blue vertical line)')),
y = 'P(X = x)'))
}
}
for (n in c(100, 500, 1000)) {
for (p in c(0.2, 0.5, 0.8)) {
x <- c(0:n)
# P(X = x) when X ~ Pois(n, p) and X <= n; lambda = n * p
y <- dpois(x, n * p)
df <- data.frame(x, y)
names(df) <- c('x', 'y')
print(df %>%
ggplot(aes(x = x, y = y)) +
geom_bar(stat = 'identity', col = 'red', fill = 'red') +
geom_line(aes(x = n * p), color = 'blue') +
labs(title = TeX(glue('The Poisson Density Function for $\\lambda$ = ', n * p,
' (n = ', n, ' and p = ', p, ')')),
subtitle = TeX(glue('$E\\[X\\] = \\lambda = $', n * p, ' (the blue vertical line)')),
y = TeX('P(X = x)')))
}
}
for (n in c(100, 200, 500)) {
for (p in c(0.1, 0.05, 0.01)) {
x <- c(0:n)
y <- dbinom(x, n, p)
z <- dpois(x, n * p)
df <- data.frame(x, y, z)
names(df) <- c('x', 'y', 'z')
print(df %>%
ggplot(aes(x = x)) +
geom_bar(aes(y = y), stat = 'identity', col = 'red', fill = 'red', alpha = 0.5) +
geom_bar(aes(y = z), stat = 'identity', col = 'blue', fill = 'blue', alpha = 0.3) +
labs(title = glue('Poisson (Red) Versus Binomial (Blue) Density Function \n for n = ', n,
' and p = ', p),
subtitle = TeX(glue('$\\lambda = n \\cdot p = $', n * p)),
y = 'P(X = x)'))
}
}
3.3 Number of Links
# exploring equations 3.1 and 3.2 on p. 76
n <- 10
M <- choose(n, 2)
p <- 0.5
x <- c(0:M)
y <- dbinom(x, M, p)
df <- data.frame(x,y)
names(df) <- c('x', 'y')
df %>%
ggplot(aes(x = x, y = y)) +
geom_point() +
geom_line(aes(x = M * p), color = 'blue') +
labs(title = TeX(glue('Equations $3.1$ and $3.2$ on p. 76 for $N = $', n, ' and $p = $', p)),
subtitle = TeX(glue('$E\\[ L \\] = N (N - 1) / 2 \\cdot p = $', n * (n - 1) * p / 2)),
x = TeX('$L$'),
y = TeX('$P_L$'))
3.4 The Degree Distribution
# exploring equation 3.7 on p. 79
n <- 10
p <- 0.5
x <- c(0:n-1)
y <- dbinom(x, n - 1, p)
df <- data.frame(x,y)
names(df) <- c('x', 'y')
df %>%
ggplot(aes(x = x, y = y)) +
geom_point() +
geom_line(aes(x = (n-1) * p), color = 'blue') +
labs(title = TeX(glue('Equation $3.7$ on p. 79 for $N = $', n, ' and $p = $', p)),
x = TeX('$k$'),
y = TeX('$p_k$'))
3.6 The Evolution of a Random Network
set.seed(42)
n <- 100
for (p in c(0.001, 0.01, 0.015, 0.02, 0.04, 0.05)) {
g <- sample_gnp(n, p)
print(ggraph(g, layout = 'circle') +
geom_edge_fan(color = 'dark blue', alpha = 0.05) +
geom_node_point(color = 'dark red', size = 2, alpha = 0.75) +
theme_graph(base_family = 'Helvetica') +
labs(title = TeX('An Example of $G(n, \\, p)$ Model'),
subtitle = glue(' n = ', n,
';\np = ', p,
';\nnumber of components = ', components(g)$no,
';\norder of the largest component = ', max(components(g)$csize),
';\naverage degree = ', n * p)))
writeLines('\n')
}
log(n) / n## [1] 0.0460517log(n)## [1] 4.60517set.seed(42)
n <- 100
p <- 0.05
g <- sample_gnp(n, p)
plot(g,
layout = layout_in_circle,
vertex.size = 2,
vertex.label = NA,
edge.lty = 3,
main = TeX(glue('An Example of $G(N, \\, p)$ Model for $n = $', n, ' and $p = $', p)))
ggraph(g, layout = 'circle') +
geom_edge_fan(color = 'dark blue', alpha = 0.25) +
geom_node_point(color = 'dark red', size = 2, alpha = 0.75) +
theme_graph(base_family = 'Helvetica') +
labs(title = TeX(glue('An Example of $G(N, \\, p)$ Model for $N = $', n, ' and $p = $', p))) 
components(g)## $membership
## [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [38] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [75] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
##
## $csize
## [1] 100
##
## $no
## [1] 1suppressMessages(df <- bind_cols(enframe(degree(g)),
enframe(transitivity(g, type = c('local')))))
df <- df %>% select(name...1, value...2, value...4)
names(df) <- c('name', 'degree', 'clustering')
x <- c(0:n)
y <- dpois(x, n * p)
df2 <- data.frame(x, y)
names(df2) <- c('x', 'y')
df %>%
ggplot(aes(x = degree, y = ..density..)) +
geom_density(fill = 'red', alpha = 0.5) +
geom_point(data = df2, aes(x = x, y = y), color = 'blue') +
labs(title = TeX('Poisson Density Function (Dots) vs Degree Distribution for $G(N, \\, p)$ (Curve)'),
subtitle = TeX(glue('$N = $', n, ' and $p = $', p)))
df %>%
ggplot(aes(x = degree, y = ..density..)) +
geom_histogram(fill = 'red', color = 'blue', binwidth = 1) +
labs(title = TeX('Histogram of Degree Distribution for $G(N, \\, p)$'),
subtitle = TeX(glue('$N = $', n, ' and $p = $', p)))
df %>%
ggplot(aes(x = clustering, y = ..density..)) +
geom_density(fill = 'red', alpha = 0.5, na.rm = TRUE) +
labs(title = TeX('KDE of Local Clustering Coefficient for $G(N, \\, p)$'),
subtitle = TeX(glue('$N = $', n, ' and $p = $', p)))
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.”