Inference on the History of a Randomly Growing Tree

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