Finding graph indices which are unbiased to network size and density is of high importance both within a given field and across fields for enhancing comparability of modern network science studies. The degree variance is an important metric for characterising network degree heterogeneity. Here, we provide an analytically valid normalisation of degree variance to replace previous normalisations which are either invalid or not applicable to all networks. It is shown that this normalisation provides equal values for graphs and their complements; it is maximal in the star graph (and its complement); and its expected value is constant with respect to density for Erdös-Rényi (ER) random graphs of the same size. We strengthen these results with model observations in ER random graphs, random geometric graphs, scale-free networks, random hierarchy networks and resting-state brain networks, showing that the proposed normalisation is generally less affected by both network size and density than previous normalisation attempts. The closed form expression proposed also benefits from high computational efficiency and straightforward mathematical analysis. Analysis of 184 real-world binary networks across different disciplines shows that normalised degree variance is not correlated with average degree and is robust to node and edge subsampling. Comparisons across subdomains of biological networks reveals greater degree heterogeneity among brain connectomes and food webs than in protein interaction networks.

中文翻译：

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标准化度方差。

在给定领域内和跨领域内，寻找与网络大小和密度无偏差的图索引对于提高现代网络科学研究的可比性具有非常重要的意义。程度方差是表征网络程度异质性的重要指标。在这里，我们提供了度方差的分析有效归一化，以替换先前的无效或不适用于所有网络的归一化。结果表明，这种归一化为图及其补码提供了相等的值。在星形图（及其补码）中最大；对于相同大小的Erdös-Rényi（ER）随机图，其期望值相对于密度是恒定的。我们通过ER随机图，随机几何图，无标度网络，随机等级网络和静止状态脑网络，表明与以前的标准化尝试相比，拟议的标准化通常受网络规模和密度的影响较小。提出的封闭形式表达式还受益于高计算效率和简单的数学分析。对184个跨不同学科的现实世界二进制网络的分析表明，归一化度方差与平均度无关，并且对节点和边缘二次采样具有鲁棒性。跨生物网络子域的比较显示，与蛋白质相互作用网络相比，大脑连接组和食物网之间的异质性更高。提出的封闭形式表达式还受益于高计算效率和简单的数学分析。对184个跨不同学科的现实世界二进制网络的分析表明，归一化度方差与平均度不相关，并且对节点和边缘二次采样具有鲁棒性。跨生物网络子域的比较显示，与蛋白质相互作用网络相比，大脑连接组和食物网之间的异质性更高。提出的闭式表达式还受益于高计算效率和简单的数学分析。对184个跨不同学科的现实世界二进制网络的分析表明，归一化度方差与平均度无关，并且对节点和边缘二次采样具有鲁棒性。跨生物网络子域的比较显示，与蛋白质相互作用网络相比，大脑连接组和食物网之间的异质性更高。