In an increasingly interconnected world, understanding and summarizing the structure of these networks becomes increasingly relevant. However, this task is nontrivial; proposed summary statistics are as diverse as the networks they describe, and a standardized hierarchy has not yet been established. In contrast, vector-valued random variables admit such a description in terms of their cumulants (e.g., mean, (co)variance, skew, kurtosis). Here, we introduce the natural analogue of cumulants for networks, building a hierarchical description based on correlations between an increasing number of connections, seamlessly incorporating additional information, such as directed edges, node attributes, and edge weights. These graph cumulants provide a principled and unifying framework for quantifying the propensity of a network to display any substructure of interest (such as cliques to measure clustering). Moreover, they give rise to a natural hierarchical family of maximum entropy models for networks (i.e., ERGMs) that do not suffer from the "degeneracy problem", a common practical pitfall of other ERGMs.

中文翻译：

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介绍图累积量：您的社交网络的方差是多少？

在一个日益互联的世界中，理解和总结这些网络的结构变得越来越重要。然而，这项任务并非易事。拟议的汇总统计与其描述的网络一样多样化，并且尚未建立标准化的层次结构。相比之下，向量值随机变量允许根据它们的累积量（例如，均值、（协）方差、偏斜、峰态）进行这种描述。在这里，我们介绍了网络累积量的自然模拟，基于越来越多的连接之间的相关性构建层次描述，无缝地结合附加信息，例如有向边、节点属性和边权重。这些图累积量提供了一个有原则和统一的框架，用于量化网络显示任何感兴趣的子结构（例如测量聚类的集团）的倾向。此外，它们为网络（即 ERGM）产生了一个最大熵模型的自然层次族，这些模型不受“退化问题”的影响，这是其他 ERGM 的一个常见的实际陷阱。