Balance theory explains how network structural configurations relate to tension in social systems, which are commonly modeled as static undirected signed graphs. We expand this modeling approach by incorporating directionality of edges and considering three levels of analysis for balance assessment: triads, subgroups, and the whole network. For triad-level balance, we develop a new measure by utilizing semicycles that satisfy the condition of transitivity. For subgroup-level balance, we propose measures of cohesiveness (intra-group solidarity) and divisiveness (inter-group antagonism) to capture balance within and among subgroups. For network-level balance, we re-purpose the normalized line index to incorporate directionality and assess balance based on the proportion of edges whose position suits balance. Through comprehensive computational analyses, we quantify, analyze, and compare patterns of social structure in triads, subgroups, and the whole network across a range of social settings. We then apply our multilevel framework to examine balance in temporal and multilayer networks to demonstrates the generalizability of our approach. In most cases, we find relatively high balance across the three levels; providing another confirmation of balance theory. We also deliver empirical evidence for the argument that balance at different levels is not the same social phenomenon measured at different scales, but represents different properties (triadic balance, internal cohesion and external division of subgroups, and overall network polarization), and should therefore be evaluated independently from one another. We propose a comprehensive yet parsimonious approach to address this need.

平衡理论解释了网络结构配置如何与社会系统中的张力相关联，社会系统通常被建模为静态无向有符号图。我们通过结合边的方向性并考虑平衡评估的三个分析级别来扩展这种建模方法：三元组、子组和整个网络。对于三元组平衡，我们通过利用满足传递性条件的半周期开发了一种新的度量。对于子组级别的平衡，我们提出了凝聚力（组内团结）和分裂（组间对抗）的度量，以捕获子组内部和子组之间的平衡。对于网络级平衡，我们重新利用归一化线索引以结合方向性并根据位置适合平衡的边的比例来评估平衡。通过全面的计算分析，我们量化、分析和比较了一系列社会环境中的三合会、亚群和整个网络中的社会结构模式。然后我们应用我们的多层次框架来检查时间和多层网络的平衡，以证明我们方法的普遍性。在大多数情况下，我们发现三个层次之间的平衡性相对较高；提供了平衡理论的另一个证实。我们还为以下论点提供了经验证据，即不同层次的平衡不是在不同尺度上衡量的同一社会现象，而是代表不同的属性（三元平衡、子群体的内部凝聚力和外部分裂以及整体网络极化），因此应该是相互独立评估。