创建有向随机图来指定幂律度分布的alpha

Question

I have a real directed graph for which I know the number of nodes and edges. The degree distribution approximates a power-law distribution. Now I want to create a random graph replicating the following features of my real graph:

• Number of nodes
• Number of edges
• (Similar) power-law indegree and out distribution

Let's assume g is my real graph of 10000 nodes and 30000 edges

exp.out = 2.2
exp.in = 2.3

set.seed(123)
g <- static.power.law.game(10000, 30000, exp.out, exp.in, multiple=TRUE)

Yet I don't know exp.out and exp.in . Then I try to estimate the power-law exponents with the plfit function ( downloaded here ):

plfit(degree(g, mode="in")+1)

# $xmin
# [1] 5
# 
# $alpha
# [1] 2.97
# 
# $D
# [1] 0.01735342

plfit(degree(g, mode="out")+1)

# $xmin
# [1] 5
# 
# $alpha
# [1] 2.83
# 
# $D
# [1] 0.01589222

From which I then derive my distribution functions (respectively for indegree and outdegree):

• p(x) ~ x^-2.97 for x >= 5
• p(x) ~ x^-2.83 for x >= 5

According to the documentation of static.power.law.game

The game simply uses static.fitness.game with appropriately constructed fitness vectors. In particular, the fitness of vertex i is i^(-alpha), where alpha = 1/(gamma-1) and gamma is the exponent given in the arguments

As far as I understand it, to replicate my alphas I should pass as gammas respectively 1.3367 (2.97=1/(x-1)) and 1.35336 (2.83=1/(x-1)). Then

set.seed(321)
random.g <- static.power.law.game(10000, 30000, 1.35336, 1.3367, multiple=TRUE)
# Error in .Call("R_igraph_static_power_law_game", no.of.nodes, no.of.edges,  : 
#                  At games.c:3748 : out-degree exponent must be >= 2, Invalid value

Yet the fact that static.power.law.game only takes degree exponents higher then or equal to 2 makes me think that probably I am missing something...

Answer 1

exp_out and exp_in should simply be the desired exponent of the out-degree and in-degree distributions, there is no need to do any transformations on the exponents you have obtained from plfit . However, note that it is unlikely that you will recover your "observed" exponents exactly due to finite size effects