This function generates a non-growing random graph with expected power-law degree distributions.
Usage
sample_fitness_pl(
no.of.nodes,
no.of.edges,
exponent.out,
exponent.in = -1,
loops = FALSE,
multiple = FALSE,
finite.size.correction = TRUE
)
Arguments
- no.of.nodes
The number of vertices in the generated graph.
- no.of.edges
The number of edges in the generated graph.
- exponent.out
Numeric scalar, the power law exponent of the degree distribution. For directed graphs, this specifies the exponent of the out-degree distribution. It must be greater than or equal to 2. If you pass
Inf
here, you will get back an Erdős-Rényi random network.- exponent.in
Numeric scalar. If negative, the generated graph will be undirected. If greater than or equal to 2, this argument specifies the exponent of the in-degree distribution. If non-negative but less than 2, an error will be generated.
- loops
Logical scalar, whether to allow loop edges in the generated graph.
- multiple
Logical scalar, whether to allow multiple edges in the generated graph.
- finite.size.correction
Logical scalar, whether to use the proposed finite size correction of Cho et al., see references below.
Details
This game generates a directed or undirected random graph where the degrees of vertices follow power-law distributions with prescribed exponents. For directed graphs, the exponents of the in- and out-degree distributions may be specified separately.
The game simply uses sample_fitness()
with appropriately
constructed fitness vectors. In particular, the fitness of vertex i is
i−α, where α=1/(γ−1)
and γ is the exponent given in the arguments.
To remove correlations between in- and out-degrees in case of directed
graphs, the in-fitness vector will be shuffled after it has been set up and
before sample_fitness()
is called.
Note that significant finite size effects may be observed for exponents smaller than 3 in the original formulation of the game. This function provides an argument that lets you remove the finite size effects by assuming that the fitness of vertex i is (i+i0−1)−α where i0 is a constant chosen appropriately to ensure that the maximum degree is less than the square root of the number of edges times the average degree; see the paper of Chung and Lu, and Cho et al for more details.
References
Goh K-I, Kahng B, Kim D: Universal behaviour of load distribution in scale-free networks. Phys Rev Lett 87(27):278701, 2001.
Chung F and Lu L: Connected components in a random graph with given degree sequences. Annals of Combinatorics 6, 125-145, 2002.
Cho YS, Kim JS, Park J, Kahng B, Kim D: Percolation transitions in scale-free networks under the Achlioptas process. Phys Rev Lett 103:135702, 2009.
See also
Random graph models (games)
erdos.renyi.game()
,
sample_()
,
sample_bipartite()
,
sample_correlated_gnp()
,
sample_correlated_gnp_pair()
,
sample_degseq()
,
sample_dot_product()
,
sample_fitness()
,
sample_forestfire()
,
sample_gnm()
,
sample_gnp()
,
sample_grg()
,
sample_growing()
,
sample_hierarchical_sbm()
,
sample_islands()
,
sample_k_regular()
,
sample_last_cit()
,
sample_pa()
,
sample_pa_age()
,
sample_pref()
,
sample_sbm()
,
sample_smallworld()
,
sample_traits_callaway()
,
sample_tree()
Author
Tamas Nepusz ntamas@gmail.com
Examples
g <- sample_fitness_pl(10000, 30000, 2.2, 2.3)
plot(degree_distribution(g, cumulative = TRUE, mode = "out"), log = "xy")