中介中心性（Centrality Measure）

在图论中，介数中心性是基于最短路径的图中中心性的度量。对于连通图中的每一对顶点，顶点之间至少存在一条最短路径，使得该路径通过的边数(对于未加权图)或边的权重之和(对于加权图) ) 最小化。每个顶点的介数中心性是通过顶点的这些最短路径的数量。

介数中心性在网络理论中得到广泛应用：它表示节点之间的程度。例如，在电信网络中，具有较高中介中心性的节点将对网络拥有更多控制权，因为更多信息将通过该节点。中介中心性被设计为中心性的一般度量：它适用于网络理论中的广泛问题，包括与社会网络、生物学、运输和科学合作相关的问题。

定义

节点 {\displaystyle v} v 的介数中心性由以下表达式给出：

$g(v)=\sum _{{s\neq v\neq t}}{\frac {\sigma _{{st}}(v)}{\sigma _{{st}}}}$

在哪里$\sigma_{st}$是从节点开始的最短路径总数$s$到节点$t$和$\sigma_{st}(v)$是通过的那些路径的数量$v$ .

请注意，节点的中介中心性与总和索引所暗示的节点对的数量成比例。因此，可以通过除以不包括的节点对数来重新调整计算$v$ ， 以便$g\in [0,1]$ .划分是由$(N-1)(N-2)$对于有向图和$(N-1)(N-2)/2$对于无向图，其中$N$是巨型组件中的节点数。请注意，这是针对可能的最高值进行缩放的，其中每条最短路径都穿过一个节点。通常情况并非如此，并且可以在不损失精度的情况下执行归一化

${\mbox{normal}}(g(v))={\frac {g(v)-\min(g)}{\max(g)-\min(g)}}$
这导致：

$\max(normal)=1$
$\min(normal)=0$
请注意，这始终是从较小范围到较大范围的缩放，因此不会丢失精度。

加权网络
在加权网络中，连接节点的链接不再被视为二元交互，而是按其容量、影响、频率等比例加权，这增加了网络内除拓扑效应之外的另一个维度的异质性。加权网络中节点的强度由其相邻边的权重之和给出。

$s_{{i}}=\sum _{{j=1}}^{{N}}a_{{ij}}w_{{ij}}$
和$a_{ij}$和$w_{ij}$节点之间的邻接矩阵和权重矩阵$i$和$j$ ， 分别。类似于在无标度网络中发现的幂律分布，给定节点的强度也遵循幂律分布。

$s(k)\approx k^{\beta }$
平均值的研究$s(b)$具有介数的顶点的强度$b$表明功能行为可以通过缩放形式来近似

$s(b)\approx b^{{\alpha }}$

以下是计算图及其各个节点的介数中心性的代码。

def betweenness_centrality(G, k=None, normalized=True, weight=None,
                           endpoints=False, seed=None):
    r"""Compute the shortest-path betweenness centrality for nodes.
  
    Betweenness centrality of a node $v$ is the sum of the
    fraction of all-pairs shortest paths that pass through $v$
  
    .. math::
  
       c_B(v) =\sum_{s,t \in V} \frac{\sigma(s, t|v)}{\sigma(s, t)}
  
    where $V$ is the set of nodes, $\sigma(s, t)$ is the number of
    shortest $(s, t)$-paths,  and $\sigma(s, t|v)$ is the number of
    those paths  passing through some  node $v$ other than $s, t$.
    If $s = t$, $\sigma(s, t) = 1$, and if $v \in {s, t}$,
    $\sigma(s, t|v) = 0$ [2]_.
  
    Parameters
    ----------
    G : graph
      A NetworkX graph.
  
    k : int, optional (default=None)
      If k is not None use k node samples to estimate betweenness.
      The value of k <= n where n is the number of nodes in the graph.
      Higher values give better approximation.
  
    normalized : bool, optional
      If True the betweenness values are normalized by `2/((n-1)(n-2))`
      for graphs, and `1/((n-1)(n-2))` for directed graphs where `n`
      is the number of nodes in G.
  
    weight : None or string, optional (default=None)
      If None, all edge weights are considered equal.
      Otherwise holds the name of the edge attribute used as weight.
  
    endpoints : bool, optional
      If True include the endpoints in the shortest path counts.
  
    Returns
    -------
    nodes : dictionary
       Dictionary of nodes with betweenness centrality as the value.
  
      
    Notes
    -----
    The algorithm is from Ulrik Brandes [1]_.
    See [4]_ for the original first published version and [2]_ for details on
    algorithms for variations and related metrics.
  
    For approximate betweenness calculations set k=#samples to use
    k nodes ("pivots") to estimate the betweenness values. For an estimate
    of the number of pivots needed see [3]_.
  
    For weighted graphs the edge weights must be greater than zero.
    Zero edge weights can produce an infinite number of equal length
    paths between pairs of nodes.
  
      
    """
    betweenness = dict.fromkeys(G, 0.0)  # b[v]=0 for v in G
    if k is None:
        nodes = G
    else:
        random.seed(seed)
        nodes = random.sample(G.nodes(), k)
    for s in nodes:
  
        # single source shortest paths
        if weight is None:  # use BFS
            S, P, sigma = _single_source_shortest_path_basic(G, s)
        else:  # use Dijkstra's algorithm
            S, P, sigma = _single_source_dijkstra_path_basic(G, s, weight)
  
        # accumulation
        if endpoints:
            betweenness = _accumulate_endpoints(betweenness, S, P, sigma, s)
        else:
            betweenness = _accumulate_basic(betweenness, S, P, sigma, s)
  
    # rescaling
    betweenness = _rescale(betweenness, len(G), normalized=normalized,
                           directed=G.is_directed(), k=k)
    return betweenness

上述函数是使用 networkx 库调用的，一旦安装了该库，您就可以最终使用它，并且必须用Python编写以下代码以实现节点的介数中心性。

>>> import networkx as nx
>>> G=nx.erdos_renyi_graph(50,0.5)
>>> b=nx.betweenness_centrality(G)
>>> print(b)

它的结果是：

{0: 0.01220586070437195, 1: 0.009125402885768874, 2: 0.010481510111098788, 3: 0.014645690907182346, 
4: 0.013407129955492722, 5: 0.008165902336070403, 6: 0.008515486873573529, 7: 0.0067362883337957575, 
8: 0.009167651113672941, 9: 0.012386122359980324, 10: 0.00711685931010503, 11: 0.01146358835858978, 
12: 0.010392276809830674, 13: 0.0071149912635190965, 14: 0.011112503660641336, 15: 0.008013362669468532,
 16: 0.01332441710128969, 17: 0.009307485134691016, 18: 0.006974541084171777, 19: 0.006534636068324543, 
20: 0.007794762718607258, 21: 0.012297442232146375, 22: 0.011081427155225095, 23: 0.018715475770172643,
 24: 0.011527827410298818, 25: 0.012294312339823964, 26: 0.008103941622217354, 27: 0.011063824792934858,
 28: 0.00876321613116331, 29: 0.01539738650994337, 30: 0.014968892689224241, 31: 0.006942569786325711,
 32: 0.01389881951343378, 33: 0.005315473883526104, 34: 0.012485048548223817, 35: 0.009147849010405877,
 36: 0.00755662592209711, 37: 0.007387027127423285, 38: 0.015993065123210606, 39: 0.0111516804297535,
 40: 0.010720274864419366, 41: 0.007769933231367805, 42: 0.009986222659285306, 43: 0.005102869708942402,
 44: 0.007652686310399397, 45: 0.017408689421606432, 46: 0.008512679806690831, 47: 0.01027761151708757, 
48: 0.008908600658162324, 49: 0.013439198921385216}

上面的结果是一个字典，描述了每个节点的介数中心性值。以上是我关于中心性度量的系列文章的延伸。继续联网！！！

参考
您可以在

https://en.wikipedia.org/wiki/Betweenness_centrality

http://networkx.readthedocs.io/en/networkx-1.10/index.html

.

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