initial = posint 

Number of initial vertices. The default value is m.

seed = integer or none

Seed for the random number generator. If an integer is specified, this is equivalent to calling randomize(seed) immediately before invoking this function. The default is none.

BarabasiAlbertGraph(n,m,opts) creates a Barabási–Albert random graph with n vertices, in which each new vertex added after the initial step is connected to m existing vertices.

The random number generator used can be seeded using the randomize function or the seed option.

The Barabási–Albert model is an algorithm for generating random graphs which are scale-free. It begins with some number of initial vertices. Additional vertices are then added incrementally until there are n vertices. Each new vertex v is connected is connected to m existing vertices with a probability which is proportional to the degree of each existing vertex at the moment v is added.

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\:$

$\mathrm{with}\left(\mathrm{RandomGraphs}\right)\:$

$G≔\mathrm{BarabasiAlbertGraph}\left(8\,3\right)$

$G≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 8\; vertices\; and\; 15\; edge(s)}$

$\mathrm{DrawGraph}\left(G\right)$

$\mathrm{DegreeSequence}\left(G\right)$

$\left[1\,4\,3\,6\,5\,5\,3\,3\right]$

$G≔\mathrm{BarabasiAlbertGraph}\left({10}^3\,4\right)$

$G≔\mathrm{Graph\; 2:\; an\; undirected\; graph\; with\; 1000\; vertices\; and\; 3984\; edge(s)}$

$\mathrm{Statistics}:-\mathrm{Histogram}\left(\mathrm{DegreeSequence}\left(G\right)\,\mathrm{discrete}\right)$

$\mathrm{graphs}≔\left[\mathrm{seq}\left(\mathrm{BarabasiAlbertGraph}\left(100\,2\,\mathrm{initial}=20\right)\,1..100\right)\right]\:$

$\mathrm{components}≔\mathrm{map}\left(\mathrm{BiconnectedComponents}\,\mathrm{graphs}\right)\:$

$\mathrm{Statistics}:-\mathrm{Histogram}\left(\mathrm{map}\left(\mathrm{numelems}\,\mathrm{components}\right)\,\mathrm{discrete}\right)$

$\mathrm{components}\left[1\right]$

$\left[\left[21\,2\right]\,\left[21\,3\,37\,18\,27\,42\,31\,5\,25\,26\,13\,30\,48\,63\,32\,55\,33\,8\,35\,9\,29\,40\,36\,23\,11\,22\,76\,93\,51\,10\,53\,56\,58\,14\,65\,52\,17\,44\,67\,69\,66\,90\,77\,81\,15\,28\,47\,59\,60\,96\,79\,89\,98\,39\,87\,34\,41\,46\,54\,72\,82\,62\,73\,74\,49\,94\,84\,45\,70\,99\,88\,64\,68\,50\,80\,85\,92\,38\,100\,71\,83\,86\,91\,75\,97\,43\,78\,7\,57\,61\,95\right]\,\left[21\,4\right]\,\left[21\,6\right]\,\left[21\,12\right]\,\left[21\,16\right]\,\left[21\,19\,24\right]\,\left[21\,20\right]\,\left[1\,21\right]\right]$

$n≔\mathrm{numelems}\left(\mathrm{components}\left[1\right]\right)$

$n≔9$

$s≔\mathrm{ColorTools}:-\mathrm{EvenSpread}\left(\mathrm{ColorTools}:-\mathrm{Color}\left(''Lab''\,''red''\right)\,n\right)$

$s≔\left[⟨Lab\; :\; 47.9\; 35.2\; -82⟩\,⟨Lab\; :\; 33.8\; 79.7\; -105⟩\,⟨Lab\; :\; 51.9\; 91\; -74.9⟩\,⟨Lab\; :\; 55.6\; 86.6\; -9.74⟩\,⟨Lab\; :\; 53.2\; 80.1\; 67.2⟩\,⟨Lab\; :\; 72.3\; 30.2\; 77.2⟩\,⟨Lab\; :\; 93.6\; -41.9\; 90.3⟩\,⟨Lab\; :\; 88.1\; -83.1\; 83.6⟩\,⟨Lab\; :\; 88.2\; -80.3\; 57.9⟩\right]$

$$\begin{gathered} \mathbf{for}i\mathbf{to}n\mathbf{do} \\ \mathrm{HighlightSubgraph}\left(\mathrm{graphs}\left[1\right]\,\mathrm{InducedSubgraph}\left(\mathrm{graphs}\left[1\right]\,\mathrm{components}\left[1\right]\left[i\right]\right)\,s\left[i\right]\,\mathrm{black}\right) \\ \mathbf{end}\mathbf{do}\: \end{gathered}$$

$\mathrm{DrawGraph}\left(\mathrm{graphs}\left[1\right]\,\mathrm{style}=\mathrm{spring}\right)$

Albert, Réka; Barabási, Albert-László (2002). "Statistical mechanics of complex networks". Reviews of Modern Physics. 74 (1): 47–97. arXiv:cond-mat/0106096. Bibcode:2002RvMP...74...47A. CiteSeerX 10.1.1.242.4753. doi:10.1103/RevModPhys.74.47. ISSN 0034-6861.

The GraphTheory[RandomGraphs][BarabasiAlbertGraph] command was introduced in Maple 2019.

For more information on Maple 2019 changes, see Updates in Maple 2019.