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| | 当连边与友谊或者合作相关时，指向的度中心常被解释为赶潮流，指出被解释为交际。对于有<math>|V|</math>个节点<math>|E|</math>条边的图<math>G:=(V,E)</math>，节点<math>v</math>的度中心定义为， | | 当连边与友谊或者合作相关时，指向的度中心常被解释为赶潮流，指出被解释为交际。对于有<math>|V|</math>个节点<math>|E|</math>条边的图<math>G:=(V,E)</math>，节点<math>v</math>的度中心定义为， |
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| − | :<math>C_{D(v)}= \deg(v)</math>
| + | <math>C_{D(v)}= \deg(v)</math> |
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| − | Calculating degree centrality for all the nodes in a graph takes [[big theta|<math>\Theta(V^2)</math>]] in a [[dense matrix|dense]] [[adjacency matrix]] representation of the graph, and for edges takes <math>\Theta(E)</math> in a [[sparse matrix]] representation.
| + | 计算图中所有节点的度中心，在密邻接矩阵表象中需要 [[big theta|<math>\Theta(V^2)</math>]], 在稀疏矩阵表象中，连边需要<math>\Theta(E)</math> 。 |
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| − | The definition of centrality on the node level can be extended to the whole graph, in which case we are speaking of ''graph centralization''.<ref>Freeman, Linton C. "Centrality in social networks conceptual clarification." Social networks 1.3 (1979): 215–239.</ref> Let <math>v*</math> be the node with highest degree centrality in <math>G</math>. Let <math>X:=(Y,Z)</math> be the <math>|Y|</math>-node connected graph that maximizes the following quantity (with <math>y*</math> being the node with highest degree centrality in <math>X</math>):
| + | 节点层面中心性的定义可以推广到整个图上，即我们说的“图中心”。<ref>Freeman, Linton C. "Centrality in social networks conceptual clarification." Social networks 1.3 (1979): 215–239.</ref> 另<math>v*</math> 表示图 <math>G</math>中度中心最大的点。 另 <math>X:=(Y,Z)</math> 为<math>|Y|</math>-与图连接使得接下来的量最大的节点(<math>y*</math> 是图<math>X</math>中心度最大的点): |
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| − | :<math>H= \sum^{|Y|}_{j=1} [C_D(y*)-C_D(y_j)]</math>
| + | <math>H= \sum^{|Y|}_{j=1} [C_D(y*)-C_D(y_j)]</math> |
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| − | Correspondingly, the degree centralization of the graph <math>G</math> is as follows:
| + | 对应的，图 <math>G</math>的度中心如下: |
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| − | :<math>C_D(G)= \frac{\sum^{|V|}_{i=1} [C_D(v*)-C_D(v_i)]}{H}</math>
| + | <math>C_D(G)= \frac{\sum^{|V|}_{i=1} [C_D(v*)-C_D(v_i)]}{H}</math> |
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| − | The value of <math>H</math> is maximized when the graph <math>X</math> contains one central node to which all other nodes are connected (a [[star graph]]), and in this case
| + | 当图<math>X</math>包含一个与其他节点都相连的中心点时 <math>H</math>的值最大 (a [[star graph]]), 此时 |
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| − | :<math>H=(n-1)\cdot((n-1)-1)=n^2-3n+2.</math>
| + | <math>H=(n-1)\cdot((n-1)-1)=n^2-3n+2.</math> |
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| − | So, for any graph <math>G:=(V,E),</math>
| + | 所以对于任意图 <math>G:=(V,E),</math> |
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| − | :<math>C_D(G)= \frac{\sum^{|V|}_{i=1} [C_D(v*)-C_D(v_i)] }{|V|^2-3|V|+2}</math>
| + | <math>C_D(G)= \frac{\sum^{|V|}_{i=1} [C_D(v*)-C_D(v_i)] }{|V|^2-3|V|+2}</math> |
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| | ===接近中心性/亲密中心性=== | | ===接近中心性/亲密中心性=== |