R: Generate scale-free graphs according to the Barabasi-Albert...

sample_pa {igraph} | R Documentation

## Generate scale-free graphs according to the Barabasi-Albert model

### Description

The BA-model is a very simple stochastic algorithm for building a graph.

### Usage

    sample_pa(
      n,
      power = 1,
      m = NULL,
      out.dist = NULL,
      out.seq = NULL,
      out.pref = FALSE,
      zero.appeal = 1,
      directed = TRUE,
      algorithm = c("psumtree", "psumtree-multiple", "bag"),
      start.graph = NULL
    )

    pa(...)

### Arguments

`n` | Number of vertices.
`power` | The power of the preferential attachment, the default is one, ie. linear preferential attachment.
`m` | Numeric constant, the number of edges to add in each time step This argument is only used if both `out.dist` and `out.seq` are omitted or NULL.
`out.dist` | Numeric vector, the distribution of the number of edges to add in each time step. This argument is only used if the `out.seq` argument is omitted or NULL.
`out.seq` | Numeric vector giving the number of edges to add in each time step. Its first element is ignored as no edges are added in the first time step.
`out.pref` | Logical, if true the total degree is used for calculating the citation probability, otherwise the in-degree is used.
`zero.appeal` | The ‘attractiveness’ of the vertices with no adjacent edges. See details below.
`directed` | Whether to create a directed graph.
`algorithm` | The algorithm to use for the graph generation. `psumtree` uses a partial prefix-sum tree to generate the graph, this algorithm can handle any `power` and `zero.appeal` values and never generates multiple edges. `psumtree-multiple` also uses a partial prefix-sum tree, but the generation of multiple edges is allowed. Before the 0.6 version igraph used this algorithm if `power` was not one, or `zero.appeal` was not one. `bag` is the algorithm that was previously (before version 0.6) used if `power` was one and `zero.appeal` was one as well. It works by putting the ids of the vertices into a bag (multiset, really), exactly as many times as their (in-)degree, plus once more. Then the required number of cited vertices are drawn from the bag, with replacement. This method might generate multiple edges. It only works if `power` and `zero.appeal` are equal one.
`start.graph` | `NULL` or an igraph graph. If a graph, then the supplied graph is used as a starting graph for the preferential attachment algorithm. The graph should have at least one vertex. If a graph is supplied here and the `out.seq` argument is not `NULL`, then it should contain the out degrees of the new vertices only, not the ones in the `start.graph`.
`...` | Passed to `sample_pa`.

### Details

This is a simple stochastic algorithm to generate a graph. It is a discrete time step model and in each time step a single vertex is added.

We start with a single vertex and no edges in the first time step. Then we add one vertex in each time step and the new vertex initiates some edges to old vertices. The probability that an old vertex is chosen is given by

P[i] ~ k[i]^alpha + a

where k[i] is the in-degree of vertex i in the current time step (more precisely the number of adjacent edges of i which were not initiated by i itself) and alpha and a are parameters given by the `power` and `zero.appeal` arguments.

The number of edges initiated in a time step is given by the `m`, `out.dist` and `out.seq` arguments. If `out.seq` is given and not NULL then it gives the number of edges to add in a vector, the first element is ignored, the second is the number of edges to add in the second time step and so on. If `out.seq` is not given or null and `out.dist` is given and not NULL then it is used as a discrete distribution to generate the number of edges in each time step. Its first element is the probability that no edges will be added, the second is the probability that one edge is added, etc. (`out.dist` does not need to sum up to one, it normalized automatically.) `out.dist` should contain non-negative numbers and at east one element should be positive.

If both `out.seq` and `out.dist` are omitted or NULL then `m` will be used, it should be a positive integer constant and `m` edges will be added in each time step.

`sample_pa` generates a directed graph by default, set `directed` to `FALSE` to generate an undirected graph. Note that even if an undirected graph is generated k[i] denotes the number of adjacent edges not initiated by the vertex itself and not the total (in- + out-) degree of the vertex, unless the `out.pref` argument is set to `TRUE`.

### Value

A graph object.

### Author(s)

Gabor Csardi csardi.gabor@gmail.com

### References

Barabasi, A.-L. and Albert R. 1999. Emergence of scaling in random networks Science, 286 509–512.

### See Also

`sample_gnp`

### Examples

    g <- sample_pa(10000)
    degree_distribution(g)

[Package igraph version 1.2.11 Index]