一般来说，degree heterogeneity 表示度的异质性，计算公式如下：
\[H = \frac{\left\langle k^2 \right\rangle}{{\left\langle k \right\rangle}^2}\]
$$$$  也就是 度的平方 的均值 比上 平均度 的平方[1].



Degree Heterogeneity [2] 中的异质性可以有很多种解释，例如Meta分析中关于异质性检验。在网络科学中，由Hu Hai-Bo 和 Wang Xiao-Fan在2008年在Physica A期刊中提出，用来描述网络的度分布的异质程度，原文：

The H considers the difference of every two degree values in a degree sequence. Actually H is equal to one-half of the relative mean difference, i.e. the arithmetic average of the absolute values of the differences between all possible pairs of node degrees: where d¯ is the average degree. Heterogeneity index provides a measure of the average degree inequality in a network,
and larger index implies higher level of heterogeneity. Apparently, $0 ≤ H < 1$. H = 0 for a completely homogeneous
network and H → 1 for a completely heterogeneous network. For any real-world networks, their H’s are all
less than 1, and H may approach one only for infinite networks. The H is superior to some other parameters in
characterizing heterogeneity, such as variance or standard deviation, since for comparing heterogeneity they demand
that two networks studied should have the same average degree.

也就是说 H 的取值范围是 [0, 1].

根据公式，Python版计算H的代码如下：

def degree_heterogeneity(G):
    '''
    @description: Calculate the degree heterogeneity of the nx.Graph() object.  
    
    Reference: H.-B. Hu and X.-F. Wang, 
    “Unified index to quantifying heterogeneity of complex networks,” 
    Physica A: Statistical Mechanics and its Applications, 
    vol. 387, no. 14, pp. 3769–3780, Jun. 2008.
    
    @param : nx.Graph()
    @return: value with range [0, 1]
    '''
    ak = np.average([n for n in dict(G.degree()).values()])
    H = sum([abs(G.degree[i]-G.degree[j]) for i in G.nodes() for j in G.nodes()]) / (2 * ak * (len(G)**2))
    return round(H, 4)

当然，也有另外一种表达：
The degree heterogeneity in directed network is defined as \(H = \frac{ E_{fo} }{\log_2 n}\) , where \(E_{fo}\) is the first-order entropy of the network [3].


参考文献：

1. Hu, Hai-Bo, and Xiao-Fan Wang. "Unified index to quantifying heterogeneity of complex networks." Physica A: Statistical Mechanics and its Applications 387.14 (2008): 3769-3780.
2. D. J. Watts and S. H. Strogatz, “Collective dynamics of ‘small-world’ networks,” Nature, vol. 393, no. 6684, pp. 440–442, Jun. 1998.
3. Zhou X, Zhang F M, Hui X B, et al. Method for measuring the heterogeneity of complex directed network based on entropy[J]. Syst. Eng, 2011, 29(8): 123-126.