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.

## Value

An igraph graph, directed or undirected.

## 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 $$i^{-\alpha}$$, where $$\alpha = 1/(\gamma-1)$$ and $$\gamma$$ 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 $$(i+i_0-1)^{-\alpha}$$ where $$i_0$$ 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.

Random graph models (games) erdos.renyi.game(), sample_(), sample_bipartite(), sample_chung_lu(), 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")