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EdgeBetweennessCentrality

EdgeBetweennessCentrality

EdgeBetweennessCentrality[g]

给出图 g 中边的中介中心性列表.

EdgeBetweennessCentrality[{vw,…}]

用规则 vw 指定图 g.

更多信息

边的中介性中心性是顶点对之间的最短路径数目.
EdgeBetweennessCentrality 作用于无向图、有向图、权重图、多冲突和混合图.

背景

EdgeBetweennessCentrality 返回一个正的机器精度数值（“边的中介中心度”）的列表，这些数值是图的边的特定中心度的近似值. 对有一条或更多边的图，边的中介中心度介于和 $2TemplateBox[{{{n, ^, 2}, /, 4}}, Floor]$ 之间（包括两端）. 边的中介中心度是对网络中边的中心性的度量，这个值基于穿过给定边的最短路径的数量. 因此它标示出了网络中对信息流至关重要的边. 这一度量在社交网络、交通运输、生物学及社会科学中均有应用.
对一个连通图，设表示顶点和之间最短路径的数量，而表示和之间经过边的最短路径的数量. 于是边的边中介中心度的定义为 .
BetweennessCentrality 用了同样的中介性的概念去求基于顶点的中介中心度.

范例

打开所有单元关闭所有单元

基本范例 (2)

计算边中介性中心性：

Wolfram Language code: g = ExampleData[{"NetworkGraph", "Friendship"}];

Wolfram Language code: EdgeBetweennessCentrality[g]

突出显示：

Wolfram Language code: coloring = {EdgeList[g], Map[ColorData["TemperatureMap"], Rescale[%]]};

Wolfram Language code: HighlightGraph[g, Style[Style @@@ Transpose[coloring], Thick]]

对边排序. 排序最前面的边在顶点对组成的多个最短路径上：

Wolfram Language code: g = ExampleData[{"NetworkGraph", "Friendship"}];

Wolfram Language code: Part[EdgeList[g], Ordering[EdgeBetweennessCentrality[g], All, Greater]]

范围 (7)

EdgeBetweennessCentrality 可用于无向图：

Wolfram Language code: EdgeBetweennessCentrality[[image]]

有向图：

Wolfram Language code: EdgeBetweennessCentrality[[image]]

权重图：

Wolfram Language code: EdgeBetweennessCentrality[[image]]

多重图：

Wolfram Language code: EdgeBetweennessCentrality[[image]]

混合图：

Wolfram Language code: EdgeBetweennessCentrality[[image]]

用规则指定图：

Wolfram Language code: EdgeBetweennessCentrality[{1 -> 3, 2 -> 1, 3 -> 6, 4 -> 6, 1 -> 5, 5 -> 4, 6 -> 1}]

EdgeBetweennessCentrality 可用于大规模图：

Wolfram Language code: g = RandomGraph[{10000, 10005}];

Wolfram Language code: EdgeBetweennessCentrality[g]//Short//Timing

应用 (5)

根据最短路径的数目对图的边进行排序：

Wolfram Language code: g = [image];

Wolfram Language code: SortBy[{EdgeList[g], EdgeBetweennessCentrality[g]}, Last]//Reverse

突出显示 CycleGraph 的边中介性中心性：

Wolfram Language code:

HighlightCentrality[g_, cc_] := HighlightGraph[g, Table[Style[EdgeList[g][[i]], ColorData["TemperatureMap"][cc[[i]] / Max[cc]]], {i, EdgeCount[g]}]];

Wolfram Language code: g = CycleGraph[8];

Wolfram Language code: cc = EdgeBetweennessCentrality[g];

Wolfram Language code: HighlightCentrality[g, cc]

Wolfram Language code: g = GridGraph[{10, 10}];

Wolfram Language code: cc = EdgeBetweennessCentrality[g];

Wolfram Language code: HighlightCentrality[g, cc]

CompleteKaryTree:

Wolfram Language code: g = CompleteKaryTree[3, 3];

Wolfram Language code: cc = EdgeBetweennessCentrality[g];

Wolfram Language code: HighlightCentrality[g, cc]

Wolfram Language code: g = PathGraph[Range[20]];

Wolfram Language code: cc = EccentricityCentrality[g];

Wolfram Language code: HighlightCentrality[g, cc]

公路网络连接芝加哥郊区. 假定芝加哥居民使用最短路径，求最中央的道路：

Wolfram Language code: g = [image];

两条路是最短路径的一部分：

Wolfram Language code: c = EdgeBetweennessCentrality[g];

Wolfram Language code: Table[EdgeList[g][[i]], {i, Ordering[c, -2]}]

表示每个的西部各州电网拓扑结构的网格网络. 确定承受最多负载的网格部分：

Wolfram Language code: g = ExampleData[{"NetworkGraph", "PowerGrid"}];

Wolfram Language code: c = EdgeBetweennessCentrality[g];

Wolfram Language code: Table[EdgeList[g][[i]], {i, Ordering[c, -2]}]

查看关键部分：

Wolfram Language code: HighlightGraph[NeighborhoodGraph[g, %, 2, GraphLayout -> "SpringEmbedding"], Style[%, Red, Thick]]

Mycobacterium bovis（牛分枝杆菌）的代谢蜂窝网络. 边中介性中心性的频率服从有一个非常尖锐的尖峰的类泊松分布：

Wolfram Language code: g = ExampleData[{"NetworkGraph", "MetabolicNetworkMycobacteriumBovis"}];

Wolfram Language code: c = EdgeBetweennessCentrality[g];

Wolfram Language code: Histogram[c]

获取最大似然参数估计：

Wolfram Language code: dist = EstimatedDistribution[Tally[c][[All, -1]], PoissonDistribution[ρ]]

属性和关系 (2)

无向图的边中介性中心性等于每个分量的中心性：

Wolfram Language code: g = GraphDisjointUnion[CycleGraph[3], CycleGraph[3], GraphLayout -> {"PackingLayout" -> "LayeredLeft"}]

Wolfram Language code: EdgeBetweennessCentrality[g]

计算每个分量的中心性给出相同的结果：

Wolfram Language code: {g1, g2} = Subgraph[g, #]& /@ ConnectedComponents[g]

Wolfram Language code: {EdgeBetweennessCentrality[g1], EdgeBetweennessCentrality[g2]}

Wolfram Language code: Join@@%

使用 EdgeIndex 获取特定顶点的中心性：

Wolfram Language code: g = ExampleData[{"NetworkGraph", "Friendship"}];

Wolfram Language code: EdgeBetweennessCentrality[g][[EdgeIndex[g, "Anna""Rudy"]]]

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