# sample_fitness_pl

##### Scale-free random graphs, from vertex fitness scores

This function generates a non-growing random graph with expected power-law degree distributions.

- Keywords
- graphs

##### 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 Erdos-Renyi 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.

##### Value

An igraph graph, directed or undirected.

##### 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.

##### Examples

```
# NOT RUN {
g <- sample_fitness_pl(10000, 30000, 2.2, 2.3)
# }
# NOT RUN {
plot(degree_distribution(g, cumulative=TRUE, mode="out"), log="xy")
# }
```

*Documentation reproduced from package igraph, version 1.2.2, License: GPL (>= 2)*