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Introducing Graph Cumulants: What is the Variance of Your Social Network?
arXiv - CS - Discrete Mathematics Pub Date : 2020-02-10 , DOI: arxiv-2002.03959 Lee M. Gunderson and Gecia Bravo-Hermsdorff
arXiv - CS - Discrete Mathematics Pub Date : 2020-02-10 , DOI: arxiv-2002.03959 Lee M. Gunderson and Gecia Bravo-Hermsdorff
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:
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。
更新日期:2020-04-15
中文翻译:
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。