Understanding the effects of the choice of the tree on the joint distribution of a tree-structured Markov random field (MRF) is crucial for fully exploiting the intelligibility of such probabilistic graphical models. Tools must be developed in this regard: this is the overarching objective of this paper. Our discussion is two-fold. First, we examine concepts specific to network centrality theory. We put forth a new conception of centrality for MRFs that not only accounts for the tree topology, but also the underlying stochastic dynamics. In this vein, we compare synecdochic pairs, random vectors comprising a MRF's component and its sum, using stochastic orders. The resulting orderings are transferred to risk-allocation quantities, which therefore serve as new centrality indices tailored to our stochastic framework. Second, we shed light on the influence of the tree's shapes, by establishing convex orderings for MRFs encrypted on trees of different shapes. This results in the design of a new partial order on tree shapes. This analysis is done within the framework of a propagation-based family of tree-structured MRFs with the uncommon property of having fixed Poisson marginal distributions unaffected by the dependence scheme. This work is a first step into the analysis of MRFs' trees' shapes and a stepping stone to extending this analysis to a broader framework.

翻译：理解树的选择对树结构马尔可夫随机场联合分布的影响，对于充分利用此类概率图模型的可解释性至关重要。必须为此开发相应工具：这是本文的核心目标。我们的讨论分为两个方面。首先，我们考察网络中心性理论特有的概念。我们提出了一种针对马尔可夫随机场的新中心性概念，它不仅考虑树的拓扑结构，还考虑了底层的随机动态。基于此，我们使用随机序比较提喻对——即包含马尔可夫随机场一个分量及其总和的随机向量。由此得到的序关系被转移到风险分配量上，从而成为适应我们随机框架的新中心性指标。其次，我们通过为在不同形状的树上编码的马尔可夫随机场建立凸序，揭示了树形状的影响。这导致了一种针对树形状的新偏序关系的设计。该分析是在一类基于传播的树结构马尔可夫随机框架内进行的，这类模型具有一个不常见的特性：其固定的泊松边缘分布不受依赖方案的影响。这项工作是分析马尔可夫随机场树形状的第一步，也是将此类分析扩展到更广泛框架的基石。