# static.power.law.game

0th

Percentile

##### 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
static.power.law.game (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
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 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.

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 static.fitness.game 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.

##### concept

Random graph model

##### 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.

##### Aliases
• static.power.law.game
##### Examples
g <- static.power.law.game(10000, 30000, 2.2, 2.3)
plot(degree.distribution(g, cumulative=TRUE, mode="out"), log="xy")
Documentation reproduced from package igraph, version 0.6.5-2, License: GPL (>= 2)

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