Proposers of the vote of thanks to Crane and Xu and contribution to the discussion of “Root and community inference on the latent growth process of a network”

We commend the authors on an illuminating article which significantly extends the state-ofthe-art in network archaeology.Notably, whereas most other literature in the area has provided methods for root inference which are theoretically grounded only in the case of random trees, the techniques in this paper have been designed to model non-tree-structured networks, as well.This has important practical implications, as many real-world networks contain one or more cycles.The main challenge faced by the probability and statistics community-for the better part of the last decade-was that it was difficult to concoct a model that was simple enough to analyze rigorously, yet complex enough to comprise a sufficiently interesting family of random graph structures.By introducing their PAPER model, Crane and Xu have proposed a model achieving an ideal balance, accompanied by a variety of attractively useful and strikingly intuitive theoretical results.The simulation results are also very impressive, in that the algorithms can be efficiently run on a single laptop even for networks with millions of nodes.The fact that the PAPER model achieves such versatility with only three parameters is a major strength of the work.This makes parameter estimation plausible (the authors suggest several methods for parameter estimation in the appendices, based on EM and Bayesian methods).From our reading, we wondered how one might decide in practice to use a more complicated model for a given network, e.g., sequential PAPER vs. non-sequential PAPER, or fixed-K vs. random K, as the more complicated networks would of course lead to more complicated inference procedures.For instance, is anything known about degree profiles of random-K networks, and are they qualitatively different from the degree profiles of fixed-K networks?(Which would be more reasonable for specific problems such as the coauthor networks studied in this paper?)We are also curious to understand better, in practice, what the confidence sets look like for K > 1 (does the algorithm output K connected components?).How might the algorithm be modified if the goal is to output a set of K-tuples such that the K-tuple containing the K root nodes is contained in this set, with a certain high probability?Section 5 of the paper presents theoretical results concerning confidence sets for the root node (or root nodes, when K > 1) of a random network.Along the lines of some of our previous work [3], we wondered how much the theory could extend to the case of constructing a confidence set for the first M > 1 nodes (say, for simplicity, in the case when K = 1).Also, can reasonable conditions be imposed under which confidence sets are proven to "persist" when the number of