Using igraph in R, I was trying to confirm that the Erdos-Reyni method of constructing random networks would indeed end up with networks with degree distributions that are well fit by a Poisson distribution. So, I ran the R the code:
library(igraph)
library(MASS) #for fitdistr function
samples<-5000
# first n is the number of nodes, the n/samples is the probability of making
# an edge between any two nodes
g <- erdos.renyi.game(samples, 20/ samples)
d <- degree(g) #finds the degree of each node in graph
# fit should be done on frequency, not on probability, and not on summary
fits<-fitdistr(d,"Poisson")
dd<- as.numeric(table(d)) # creates a summary of each degree and its frequency
plot(dd, xlab = "degree", ylab="frequency") # plot degree distribution
a<-1:length(dd)
# multiply by original samples to create frequency plots
lines(a,dpois(a,lambda=fits$estimate)* samples)
I get a plot in which it looks like the distribution of degrees is indeed Poisson but the best fitting Poisson distribution is dramatically right shifted compared to the actual degree distribution. Given that the graph was created so that there would be on average 20 links per node, it makes sense that the best fitting lambda value is around 20, but why does the actual distribution have a mode around 14 (although the mean of the degrees is also around 20)?