barabasi.game

0th

Percentile

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

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

Keywords
graphs
Usage
barabasi.game(n, power = 1, m = NULL, out.dist = NULL, out.seq = NULL, 
    out.pref = FALSE, zero.appeal = 1, directed = TRUE, time.window = NULL)
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.
time.window
Integer constant, if given only edges added in the previous time.window time steps are counted as the basis of preferential attachment.
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 $$P[i] \sim k_i^\alpha+a$$ 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.

barabasi.game 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.

If the time.window argument is not NULL then $k_i$ is the number of adjacent edges of $i$ added in the previous time.window time steps.

Note that barabasi.game might generate graphs with multiple edges.

Value

  • A graph object.

concept

  • Preferential attachment model
  • Random graph model

References

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

See Also

random.graph.game

Aliases
  • barabasi.game
  • ba.game
Examples
g <- barabasi.game(10000)
degree.distribution(g)
Documentation reproduced from package igraph, version 0.5.1, License: GPL (>= 2)

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