The spread of infectious disease in a human community or the proliferation of fake news on social media can be modelled as a randomly growing tree-shaped graph. The history of the random growth process is often unobserved but contains important information such as the source of the infection. We consider the problem of statistical inference on aspects of the latent history using only a single snapshot of the final tree. Our approach is to apply random labels to the observed unlabelled tree and analyse the resulting distribution of the growth process, conditional on the final outcome. We show that this conditional distribution is tractable under a shape exchangeability condition, which we introduce here, and that this condition is satisfied for many popular models for randomly growing trees such as uniform attachment, linear preferential attachment and uniform attachment on a D-regular tree. For inference of the root under shape exchangeability, we propose O(n log n) time algorithms for constructing confidence sets with valid frequentist coverage as well as bounds on the expected size of the confidence sets. We also provide efficient sampling algorithms which extend our methods to a wide class of inference problems.

中文翻译：

---

对随机生长树历史的推断

传染病在人类社区中的传播或社交媒体上假新闻的扩散可以建模为一个随机增长的树形图。随机生长过程的历史通常未被观察到，但包含重要信息，例如感染源。我们仅使用最终树的单个快照来考虑对潜在历史方面的统计推断问题。我们的方法是将随机标签应用于观察到的未标记树，并以最终结果为条件分析生长过程的结果分布。我们表明这种条件分布在形状可交换性下是易于处理的我们在此介绍的条件，并且该条件满足许多流行的随机生长树模型，例如D正则树上的均匀附着、线性优先附着和均匀附着。为了推断形状可交换性下的根，我们提出了O ( n  log  n ) 时间算法来构建具有有效频率覆盖范围以及置信集预期大小范围的置信集。我们还提供有效的采样算法，将我们的方法扩展到广泛的推理问题。