输入以适应网络的程度分布的幂律

Question

I would like to use R to test whether the degree distribution of a network behaves like a power-law with scale-free property. Nonetheless, I've read different people doing this in many different ways, and one confusing point is the input one should use in the model.

Barabasi, for example, recommends fitting a power-law to the 'complementary cumulative distribution' of degrees ( see Advanced Topic 3.B of chapter 4, figure 4.22 ). However, I've seen people fit a power-law to the degrees of the graph (obtained with igraph::degree(g) ), and I've also seen others fitting a power-law to a degree distribution, obtained via igraph::degree_distribution(g, cumulative = T)

As you can see in the reproducible example below, these options give very different results. Which one is correct? and how can I get the "complementary cumulative distribution of degrees" to from a graph so I can fit a power-law?

library(igraph)

# create a graph
  set.seed(202)
  g <- static.power.law.game(500, 1000, exponent.out= 2.2, exponent.in = 2.2, loops = FALSE, multiple = T)

# get input to fit power-law.

  # 1) degrees of the nodes
    d <- degree(g, v = V(g), mode ="all")
    d <- d[ d > 0] # remove nodes with no connection

  # OR ?

  # 2) cumulative degree distribution
    d <- degree_distribution(g, mode ="all", cumulative = T)

# Fit power law
  fit <- fit_power_law(d, impelementation = "R.mle")

Answer 1

Well, the problem here is that you have 2 different statistics here. The degree of a node shows how many connections it has to other nodes. The degree distribution is the probability distribution of those degrees over the network.

For me it doesn't make much sense to apply the igraph::fit_power_law on a degree distribution as the degree distribution is already a power law to a certain extent.

However, don't forget that the igraph::fit_power_law has more options than the implementation argument, which will result in different things, depending on what you're "feeding it".