Estimated Poisson Distribution

Question

I have a graph, and I calculated the degree distribution and degree as follows:

library(igraph) # for these two functions

dd <- degree_distribution(graph) 
d <- degree(graph)

From this, I evaluated the law of power to find out if my distribution complies with the "Law of Power":

degree = 1:max(d)
probability = dd[-1]

nonzero.position = which(probability != 0)
probability = probability[nonzero.position]
degree = degree[nonzero.position]

reg = lm(log(probability) ~ log(degree))
cozf = coef(reg)

power.law.fit = function(x) exp(cozf[[1]] + cozf[[2]] * log(x))

From this I built points and a power law using ggplot2. The result in the following image:

df <- data.frame(x = degree, y = probability)
  print(
      ggplot(df, aes(x,y,colour="Distribuição"))+
        geom_point(shape = 4) +
        stat_function(fun = power.law.fit, geom = "line", aes(colour="Power Law"))+

        labs(title = "Grafo", subtitle = "Distribuição dos Graus",
             x="K", y="P(k)", colour="Legenda")+
        scale_color_brewer(palette="Dark2")
  )

As you can see, my distribution does not match Power Law! I would like to evaluate the Poisson distribution and the graph in one graph. Despite the fact that I am not sure that my distribution should not (or should not) be Poisson, I would like to combine it with Power Law. I have no idea how to estimate this distribution (Poisson) from the data and calculate the average degree.

Can anyone help me?

, , (700 ), . .

Answer 1

?dpois:

p (x) = λ ^ x exp (-λ)/x!

x = 0, 1, 2,.... E (X) = Var (X) = λ.

, :

mysecret <- ####

x <- data.frame(xes = rpois(50, mysecret))

> x$xes
 [1] 0 2 2 1 1 4 1 1 0 2 2 2 1 0 0 1 2 3 2 4 2 1 0 3 2 1 3 1 2 1 5 0 2 3 2 1 0 1 2 3 0 1 2 2 0 3 2 2 2 3


> mean(x$xes)
[1] 1.66
> var(x$xes)
[1] 1.371837

, - 1.66 1.37. :

library(ggplot2)
ggplot(x, aes(xes)) + 
  geom_histogram(aes(y = ..density.., color = "Raw data"), 
                 fill = "white", binwidth = 1, center = 0, size = 1.5) +
  stat_summary(fun.y = dpois, aes(x = xes, y = xes, color = "Density based on E(X)"), 
               fun.args = list(lambda = 1.66), geom = "line", size = 1.5) +
  stat_summary(fun.y = dpois, aes(x = xes, y = xes, color = "Density based on Var(X)"), 
               fun.args = list(lambda = 1.37), geom = "line", size = 1.5)

. stat_function geom_density , . , , .

, MASS:

MASS::fitdistr(x$xes, dpois, start = list(lambda = 1))

    lambda  
  1.6601563 
 (0.1822258)

, :

library(dplyr)
df <- data_frame(xes = seq.int(max(x$xes)+1)-1,
                 dens.m = dpois(xes, 1.66),
                 dens.u = dpois(xes, 1.66+0.18),
                 dens.l = dpois(xes, 1.66-0.18))

> df
# A tibble: 6 x 4
    xes     dens.m     dens.u     dens.l
  <dbl>      <dbl>      <dbl>      <dbl>
1     0 0.19013898 0.15881743 0.22763769
2     1 0.31563071 0.29222406 0.33690378
3     2 0.26197349 0.26884614 0.24930880
4     3 0.14495866 0.16489230 0.12299234
5     4 0.06015785 0.07585046 0.04550717
6     5 0.01997240 0.02791297 0.01347012

ggplot(x, aes(xes)) + 
  geom_histogram(aes(y = ..density..), color = "black",
                 fill = "white", binwidth = 1, center = 0, size = 1.5) +
  geom_ribbon(data = df, aes(xes, ymin = dens.l, ymax = dens.u), fill = "grey50", alpha = 0.5) +
  geom_line(data = df, aes(xes, dens.m, color = "Based on E(X)\n+/-1 SD of lambda"), size = 1.5)

, , , λ = 1,66 +/- 0,18.

, 1,5.