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
\(i^{-\alpha}\), 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.

## See also

Random graph models (games)
`erdos.renyi.game()`

,
`sample_bipartite()`

,
`sample_correlated_gnp_pair()`

,
`sample_correlated_gnp()`

,
`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_age()`

,
`sample_pa()`

,
`sample_pref()`

,
`sample_sbm()`

,
`sample_smallworld()`

,
`sample_traits_callaway()`

,
`sample_tree()`

,
`sample_()`

## 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")
```