Generating a connected and directed network in R

Question

I am trying to generate a directed and connected scale-free network in R, such that, the in- and out-degrees of the network follow a power-law. I tried the below methods,

1. g<-sample_fitness_pl(10000, 1000000, 2.7, 2.7) generates a network with a in-/out-degree distribution that does not follow a power-law

2. The following function generates power-law distributions of the in- and out-degrees of the nodes, however, the resulting network is not even weakly connected,

N <- 10000
    
g <- sample_fitness(10*N, sample((1:50)^-2, N, replace=TRUE),sample((1:50)^-2, N, replace=TRUE))

I appreciate any comments that help me learn more about it.

Answer 1

As explained in the documentation, the function sample_fitness_pl() generates a non-growing random graph with expected power-law degree distributions. We can verify this by plotting the (in-) degree distribution and fitting a power-law distribution function f(x) = c.x^(-α) with non-linear least-square (nls) model.

library(igraph)
g<-sample_fitness_pl(10000, 1000000, 2.7, 2.7)
df <- as.data.frame(table(degree(g, mode='in')))
names(df) <- c('in.deg', 'freq')
df$in.deg <- as.integer(as.character(df$in.deg))
head(df)
#   in.deg freq
# 1     31    3
# 2     32    3
# 3     33    3
# 4     34    8
# 5     35    9
# 6     36   14
model <- nls(freq ~ c*in.deg^(-alpha), data=df, start = list(c = 10,alpha = 0.1))
summary(model)
# Formula: freq ~ c * in.deg^(-alpha)

# Parameters:
#        Estimate Std. Error t value Pr(>|t|)    
# c     3.281e+03  8.315e+02   3.946 9.12e-05 ***
# alpha 9.471e-01  5.926e-02  15.981  < 2e-16 ***
# ---
# Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

# Residual standard error: 31.54 on 487 degrees of freedom

# Number of iterations to convergence: 35 
# Achieved convergence tolerance: 8.192e-06

which shows that the degree distribution can be approximated by the power-law distribution f(x) = 3281 . x ^(-0.95), as shown below (the red curve represents the fitted distribution):

deg <- seq(min(df$in.deg), max(df$in.deg), 1)
pow.law.pred <- predict(model, newdata = data.frame(in.deg=deg))
pred.df <- data.frame(deg=deg, pred=pow.law.pred)
library(ggplot2)
ggplot(df, aes(in.deg, freq)) + geom_point() + 
  geom_line(data=pred.df, aes(deg, pred), col='red')

Also, the log-log plot below shows resemblance to power-law distribution, except possibly a few nodes with low in-degree values (we can remove them?)

ggplot(df, aes(in.deg, freq)) + geom_point() +   coord_trans(y ='log10', x='log10')

Also, we can find that the graph is connected, since the number of components is 1

count_components(g) 
[1] 1