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BarabasiAlbertGraphDistribution[n,k]

表示 n 个顶点的图的 Barabasi–Albert 图分布，其中每个步骤添加一个具有 k 条边的新顶点.

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范例  
基本范例  
范围  
应用  
属性和关系  
巧妙范例  
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BarabasiAlbertGraphDistribution

BarabasiAlbertGraphDistribution[n,k]

表示 n 个顶点的图的 Barabasi–Albert 图分布，其中每个步骤添加一个具有 k 条边的新顶点.

更多信息

范例

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基本范例  (2)

生成一个伪随机图：

Wolfram Language code: RandomGraph[BarabasiAlbertGraphDistribution[30, 2]]

度分布：

Wolfram Language code: RandomGraph[BarabasiAlbertGraphDistribution[10 ^ 4, 2]];
Wolfram Language code: EmpiricalDistribution[VertexDegree[%]]

概率密度函数：

Wolfram Language code: DiscretePlot[PDF[%, k], {k, 7, 30}]

范围  (3)

生成简单无向图：

Wolfram Language code: RandomGraph[BarabasiAlbertGraphDistribution[10, 3]]

生成一组伪随机图：

Wolfram Language code: RandomGraph[BarabasiAlbertGraphDistribution[10, 3], 4]

计算概率和统计属性：

Wolfram Language code: 𝒟 = GraphPropertyDistribution[VertexDegree[g, 1], gBarabasiAlbertGraphDistribution[30, 4]];
Wolfram Language code: NProbability[x ≥ 10, x𝒟]

应用  (3)

自治系统层级上的互联网可以使用 BarabasiAlbertGraphDistribution 建模：

Wolfram Language code: g = ExampleData[{"NetworkGraph", "Internet"}];
Wolfram Language code: 𝒢 = BarabasiAlbertGraphDistribution[VertexCount[g], Round[EdgeCount[g] / VertexCount[g]]]

该模型捕捉了经验度分布的幂律特性：

Wolfram Language code: {Histogram[VertexDegree[g], {"Log", 10}, {"Log", "PDF"}], Histogram[VertexDegree[RandomGraph[𝒢]], {"Log", 10}, {"Log", "PDF"}]}

该模型具有较低的集群系数：

Wolfram Language code: N[GlobalClusteringCoefficient[RandomGraph[𝒢]]]
Wolfram Language code: N[GlobalClusteringCoefficient[g]]

使用 Barabasi–Albert图分布作为 Western States Power Grid 网络的模型：

Wolfram Language code: g = ExampleData[{"NetworkGraph", "PowerGrid"}];
Wolfram Language code: 𝒢 = BarabasiAlbertGraphDistribution[VertexCount[g], Round[EdgeCount[g] / VertexCount[g]]]

该模型捕捉了经验度分布的幂律特性：

Wolfram Language code: f[g_] := Map[{#[[1]], #[[2]] / VertexCount[g]}&, Tally[VertexDegree[g]]];
Wolfram Language code: {ListLogLogPlot[f[g]], ListLogLogPlot[f[RandomGraph[𝒢]]]}

一个具有400个人和突出的核心的社交网络使用 BarabasiAlbertGraphDistribution 建模. 求核心处的人与网络中最遥远的人之间的预期节点（ties）数目：

Wolfram Language code: 𝒟 = GraphPropertyDistribution[VertexEccentricity[g, First[GraphHub[g]]], gBarabasiAlbertGraphDistribution[400, 3]];
Wolfram Language code: NExpectation[x, x𝒟]

属性和关系  (5)

顶点数目的分布：

Wolfram Language code: GraphPropertyDistribution[VertexCount[g], gBarabasiAlbertGraphDistribution[n, k]]

边数目的分布：

Wolfram Language code: GraphPropertyDistribution[EdgeCount[g], gBarabasiAlbertGraphDistribution[n, k]]

度分布：

Wolfram Language code: 𝒟[n_, k_] := EmpiricalDistribution[VertexDegree[RandomGraph[BarabasiAlbertGraphDistribution[n, k]]]];

该分布可以使用 ZipfDistribution 近似：

Wolfram Language code: ℰ = TruncatedDistribution[{5, ∞}, ZipfDistribution[2]];
Wolfram Language code: DiscretePlot[Evaluate[{PDF[𝒟[10 ^ 4, 6], d], PDF[ℰ, d]}], {d, 7, 25}, Joined -> True]

度分布服从幂律：

Wolfram Language code: PDF[ℰ, d]

使用 RandomSample 模拟 BarabasiAlbertGraphDistribution：

Wolfram Language code: barabasi[n_, k_] /; n ≤ k + 1 := CompleteGraph[n]
Wolfram Language code: barabasi[n_, k_] /; n > k + 1 := Module[{g = barabasi[n - 1, k]}, Graph[Join[EdgeList[g], Map[n#&, RandomSample[VertexDegree[g] -> VertexList[g], k]]]]]

伪随机图：

Wolfram Language code: Table[barabasi[n, 2], {n, 4, 7}]

在 BarabasiAlbertGraphDistribution[n,k] 中，有尺寸为 k+1 的最大团：

Wolfram Language code: RandomGraph[BarabasiAlbertGraphDistribution[8, 3]]
Wolfram Language code: FindClique[%]

巧妙范例  (1)

随机着色的顶点：

Wolfram Language code: HighlightGraph[RandomGraph[BarabasiAlbertGraphDistribution[10 ^ 2, 3]], Table[Style[i, RandomChoice[ColorData[45, "ColorList"]]], {i, 10 ^ 2}]]
Wolfram Research (2010)，BarabasiAlbertGraphDistribution，Wolfram 语言函数，https://reference.wolfram.com/language/ref/BarabasiAlbertGraphDistribution.html.

文本

Wolfram Research (2010)，BarabasiAlbertGraphDistribution，Wolfram 语言函数，https://reference.wolfram.com/language/ref/BarabasiAlbertGraphDistribution.html.

CMS

Wolfram 语言. 2010. "BarabasiAlbertGraphDistribution." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/BarabasiAlbertGraphDistribution.html.

APA

Wolfram 语言. (2010). BarabasiAlbertGraphDistribution. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/BarabasiAlbertGraphDistribution.html 年

BibTeX

@misc{reference.wolfram_2026_barabasialbertgraphdistribution, author="Wolfram Research", title="{BarabasiAlbertGraphDistribution}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/BarabasiAlbertGraphDistribution.html}", note=[Accessed: 09-September-2026]}

BibLaTeX

@online{reference.wolfram_2026_barabasialbertgraphdistribution, organization={Wolfram Research}, title={BarabasiAlbertGraphDistribution}, year={2010}, url={https://reference.wolfram.com/language/ref/BarabasiAlbertGraphDistribution.html}, note=[Accessed: 09-September-2026]}