Chap. 5: The Barabasi-Albert Model
# configuring figure size
options(repr.plot.width = 6, repr.plot.height = 4)
Loading Packages
# uncomment the following lines if you have not installed qdread/forestscaling; for more information, see https://github.com/qdread/forestscaling/
#install.packages('remotes')
#remotes::install_github('qdread/forestscaling')
library(tidyverse)
library(igraph)
library(igraphdata)
library(ggraph)
library(e1071)
library(forestscaling)
library(glue)
library(latex2exp)
5.3 The Barabasi-Albert Model
set.seed(42)
# the order of the graph
n <- 10
# the number of edges to be added at each step
for (m in c(2, 4, 8, 16, 32)) {
g <- sample_pa(n, m = m, directed = FALSE, algorithm = 'psumtree')
plot(g,
layout = layout_in_circle,
vertex.size = 2,
vertex.label = NA,
edge.lty = 3,
main = 'An Example of Barabasi-Albert Model',
sub = glue('with ', n, ' Nodes and ', m, ' Edges Added at Each Step'))
writeLines('\n')
}
set.seed(42)
n <- 100
for (m in c(2, 4, 8, 16, 32)) {
g <- sample_pa(n, m = m, directed = FALSE, algorithm = 'psumtree')
plot(g,
layout = layout_in_circle,
vertex.size = 2,
vertex.label = NA,
edge.lty = 3,
main = 'An Example of Barabasi-Albert Model',
sub = glue('with ', n, ' Nodes and ', m, ' Edges Added at Each Step'))
writeLines('\n')
}
set.seed(42)
n <- 100
DF <- NULL
for (m in c(2, 4, 8, 16, 32)) {
g <- sample_pa(n, m = m, directed = FALSE, algorithm = 'psumtree')
# print(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) +
# theme_graph(base_family = 'Helvetica') +
# labs(title = 'An Example of Barabasi-Albert Model',
# subtitle = glue('with ', n, ' Nodes and ', m, ' Edges Added at Each Step')))
# writeLines('\n')
suppressMessages(df <- bind_cols(enframe(eccentricity(g)),
enframe(betweenness(g)),
enframe(degree(g)),
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', 'degree', 'clustering')
DF2 <- df %>%
summarize(m = m,
avg_deg = mean(degree),
delta = max(degree),
diam = max(eccentricity),
radius = min(eccentricity),
avg_cc = mean(clustering, na.rm = TRUE),
exponent = fit_power_law(degree, implementation = 'plfit')$alpha)
DF <- rbind(DF, DF2)
}
DF
set.seed(42)
n <- 1000
DF <- NULL
for (m in c(2, 4, 8, 16, 32)) {
g <- sample_pa(n, m = m, directed = FALSE, algorithm = 'psumtree')
# print(ggraph(g, layout = 'circle') +
# geom_edge_fan(edge_linetype = 3, color = 'dark blue', alpha = 0.25) +
# geom_node_point(color = 'dark red', size = 2, alpha = 0.75) +
# theme_graph(base_family = 'Helvetica') +
# labs(title = 'An Example of Barabasi-Albert Model',
# subtitle = glue('with ', n, ' Nodes and ', m, ' Edges Added at Each Step')))
# writeLines('\n')
suppressMessages(df <- bind_cols(enframe(eccentricity(g)),
enframe(betweenness(g)),
enframe(degree(g)),
enframe(transitivity(g, type = c('local')))))
df <- df %>% select(name...1, value...2, value...4, value...6, value...8)
names(df) <- c('name', 'eccentricity', 'betweenness', 'degree', 'clustering')
DF2 <- df %>%
summarize(m = m,
avg_deg = mean(degree),
delta = max(degree),
diam = max(eccentricity),
radius = min(eccentricity),
avg_cc = mean(clustering, na.rm = TRUE),
exponent = fit_power_law(degree, implementation = 'plfit')$alpha)
DF <- rbind(DF, DF2)
}
DF
set.seed(42)
n <- 1000
for (m in c(2, 4, 8, 16, 32)) {
g <- sample_pa(n, m = m, directed = FALSE, algorithm = 'psumtree')
df <- enframe(degree(g))
df <- df %>% select(name, value)
names(df) <- c('name', 'degree')
exponent <- fit_power_law(df$degree, implementation = 'plfit')$alpha
df2 <- logbin(df$degree, n = 100)
total2 <- sum(df2$bin_value)
print(df2 %>%
filter(bin_count != 0) %>%
ggplot(aes(x = bin_midpoint, y = bin_value / total2)) +
geom_point(color = 'blue', size = .5) +
geom_line(data = df,
aes(x = degree, y = 2 * m ** (exponent - 1) * (degree) ** (-exponent)),
color = 'red') +
geom_line(data = df,
aes(x = degree, y = 2 * m ** 2 * (degree) **(-3)),
color = 'yellow') +
scale_x_log10() +
scale_y_log10() +
labs(y = TeX('$p_k$')) +
labs(title = 'Log-Log Scale, Linear Binning for Degree Distribution',
subtitle = glue('with Fitted Power Law Curves (Eq. 5.9: Red; Eq. 5.15: Yellow) in a Barabasi-Albert Graph \nwith ', n, ' Nodes and ', m, ' Edges Added at Each Step')))
writeLines('\n')
}
5.5 Degree Distribution
set.seed(42)
n <- 100
for (power in c(0, 0.5, 1, 1.5, 2, 2.5, 3)) {
g <- sample_pa(n, power = power, directed = FALSE, algorithm = 'psumtree')
plot(g,
layout = layout_in_circle,
vertex.size = 2,
vertex.label = NA,
edge.lty = 3,
main = 'An Example of Barabasi-Albert Model',
sub = glue('with ', n, ' Nodes and the Power of Preferential Attachment of ', power))
writeLines('\n')
}
set.seed(42)
n <- 100
DF <- NULL
# the power of the preferential attachment
for (power in c(0, 0.5, 1, 1.5, 2, 2.5, 3)) {
m <- 8
g <- sample_pa(n, m = m, power = power, directed = FALSE, algorithm = 'psumtree')
# print(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) +
# theme_graph(base_family = 'Helvetica') +
# labs(title = 'An Example of Barabasi-Albert Model',
# subtitle = glue('with ', n, ' Nodes, ', m, ' Edges Added at Each Step, \nand the Power of Preferential Attachment of ', power)))
# writeLines('\n')
suppressMessages(df <- bind_cols(enframe(eccentricity(g)),
enframe(betweenness(g)),
enframe(degree(g)),
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', 'degree', 'clustering')
DF2 <- df %>%
summarize(power = power,
avg_deg = mean(degree),
delta = max(degree),
diam = max(eccentricity),
radius = min(eccentricity),
avg_cc = mean(clustering, na.rm = TRUE),
exponent = fit_power_law(degree, implementation = 'plfit')$alpha)
DF <- rbind(DF, DF2)
}
DF
set.seed(42)
n <- 500
DF <- NULL
for (power in c(0, 0.5, 1, 1.5, 2, 2.5, 3)) {
m <- 8
g <- sample_pa(n, m = m , power = power, directed = FALSE, algorithm = 'psumtree')
# print(ggraph(g, layout = 'circle') +
# geom_edge_fan(edge_linetype = 3, color = 'dark blue', alpha = 0.25) +
# geom_node_point(color = 'dark red', size = 2, alpha = 0.75) +
# theme_graph(base_family = 'Helvetica') +
# labs(title = 'An Example of Barabasi-Albert Model',
# subtitle = glue('with ', n, ' Nodes, ', m, ' Edges Added at Each Step, \nand the Power of Preferential Attachment of ', power)))
# writeLines('\n')
suppressMessages(df <- bind_cols(enframe(eccentricity(g)),
enframe(betweenness(g)),
enframe(degree(g)),
enframe(transitivity(g, type = c('local')))))
df <- df %>% select(name...1, value...2, value...4, value...6, value...8)
names(df) <- c('name', 'eccentricity', 'betweenness', 'degree', 'clustering')
DF2 <- df %>%
summarize(power = power,
avg_deg = mean(degree),
delta = max(degree),
diam = max(eccentricity),
radius = min(eccentricity),
avg_cc = mean(clustering, na.rm = TRUE),
exponent = fit_power_law(degree, implementation = 'plfit')$alpha)
DF <- rbind(DF, DF2)
}
DF
set.seed(42)
n <- 1000
for (power in c(0, 0.5, 1, 1.5, 2, 2.5, 3)) {
m <- 8
g <- sample_pa(n, m = m, power = power, directed = FALSE, algorithm = 'psumtree')
df <- enframe(degree(g))
df <- df %>% select(name, value)
names(df) <- c('name', 'degree')
exponent <- fit_power_law(df$degree, implementation = 'plfit')$alpha
df2 <- logbin(df$degree, n = 100)
total2 <- sum(df2$bin_value)
print(df2 %>%
filter(bin_count != 0) %>%
ggplot(aes(x = bin_midpoint, y = bin_value / total2)) +
geom_point(color = 'blue', size = .5) +
geom_line(data = df,
aes(x = degree, y = 2 * m ** (exponent - 1) * (degree) ** (-exponent)),
color = 'red') +
geom_line(data = df,
aes(x = degree, y = 2 * m ** 2 * (degree) **(-3)),
color = 'yellow') +
scale_x_log10() +
scale_y_log10() +
labs(y = TeX('$p_k$'), x = 'k') +
labs(title = 'Log-Log Scale, Linear Binning for Degree Distribution',
subtitle = glue('with Fitted Power Law Curves (Eq. 5.9: Red; Eq. 5.15: Yellow) in a Barabasi-Albert Graph \nwith ', n, ' Nodes, ', m, ' Edges Added at Each Step, \nand the Power of Preferential Attachment of ', power)))
writeLines('\n')
}
