With the increasing availability of behavioral data from diverse digital sources, such as social media sites and cell phones, it is now possible to obtain detailed information about the structure, strength, and directionality of social interactions in varied settings. While most metrics of network structure have traditionally been defined for unweighted and undirected networks only, the richness of current network data calls for extending these metrics to weighted and directed networks. One fundamental metric in social networks is edge overlap, the proportion of friends shared by two connected individuals. Here, we extend definitions of edge overlap to weighted and directed networks and present closed-form expressions for the mean and variance of each version for the Erdős–Rényi random graph and its weighted and directed counterparts. We apply these results to social network data collected in rural villages in southern Karnataka, India. We use our analytical results to quantify the extent to which the average overlap of the empirical social network deviates from that of corresponding random graphs and compare the values of overlap across networks. Our novel definitions allow the calculation of edge overlap for more complex networks, and our derivations provide a statistically rigorous way for comparing edge overlap across networks.

中文翻译：

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加权和定向社交网络中的边缘重叠

随着来自不同数字来源（例如社交媒体网站和手机）的行为数据的可用性越来越高，现在可以获得有关不同环境中社交互动的结构、强度和方向性的详细信息。虽然网络结构的大多数指标传统上仅针对未加权和无向网络定义，但当前网络数据的丰富性要求将这些指标扩展到加权和有向网络。社交网络中的一个基本指标是边缘重叠，即两个相互关联的个人共享的朋友比例。在这里，我们将边缘重叠的定义扩展到加权和有向网络，并为 Erdős-Rényi 随机图及其加权和有向对应物的每个版本的均值和方差提供封闭形式的表达式。我们将这些结果应用于在印度卡纳塔克邦南部农村收集的社交网络数据。我们使用我们的分析结果来量化经验社交网络的平均重叠与相应随机图的平均重叠程度，并比较跨网络的重叠值。我们的新颖定义允许计算更复杂网络的边缘重叠，我们的推导提供了一种统计上严格的方法来比较网络之间的边缘重叠。