Filippo Ascolani, Antonio Lijoi and Igor Prünster’s contribution to the Discussion of ‘Root and community inference on the latent growth process of a network’ by Crane and Xu

原文摘要

We congratulate the authors for their timely and insightful contribution, which introduces a novel Bayesian approach to inferring the latent structure (early history and community detection) given a current observation of a network with n nodes. The Bayesian component of the model is summarized through a uniform prior on the random relabelling associated to the observed network. The methodological innovation is remarkably complemented by strong theoretical (frequentist) guarantees concerning uncertainty quantification. The sequential formulation of the preferential attachment component of the model exhibits many interesting connections with the Bayesian nonparametric literature. Indeed, the mechanism for sampling new edges is reminiscent of Pólya urns: the colours of the balls in the urn are the nodes labelled by the community they belong to. Moreover, in the random K roots model, the authors consider the case where K increases with n and the probability of creating a new root is the same of sampling a new value from a Dirichlet process (Ferguson, 1973) with concentration parameter α0/(2β+α)⁠; each tree corresponds to a different cluster of nodes. Within the Dirichlet process framework, the probability of generating a new tree depends only on the number of nodes in the forest, whereas the number of existing trees does not have any impact. This generative scheme might be restrictive in many settings and it is often desirable for the probability of creating a new tree to explicitly depend on the number of existing trees. In Bayesian nonparametrics, the latter requirement corresponds to prediction rules arising from the class of Gibbs-type priors (De Blasi et al., 2015): the most popular instances are the Pitman–Yor (Pitman & Yor, 1997) and the normalized generalized gamma (Lijoi et al., 2007) processes, which both generalize the Dirichlet process and lead to an asymptotic growth of the number of trees of order nσ⁠, with σ∈(0,1)⁠. What would the impact of different predictive structures on the model properties be? Even more flexible behaviours can be obtained through hierarchical compositions (Camerlenghi et al., 2019, 2018), which do not lead to Gibbs-type priors. For instance, Dirichlet process hierarchies lead to iterated logarithmic behaviours, with the number of iterations equal to the number of hierarchies. This growth rate, which can be made as slow as desired but still leads to an infinite number of trees, might have noteworthy implications. Finally, the paper addresses the case of a single network. However, in many situations one may face distinct networks with no common nodes but likely similar features: for example different academic fields may share similar co-authorship structures. Therefore being able to borrow information across different networks would often be beneficial leveraging some suitable form of probabilistic symmetry. There have been some recent proposals of partially exchangeable models for stochastic block models (see, e.g. Durante et al., 2023) that allow for prediction of the clustering of future nodes through a probabilistically coeherent sequential procedure in a spirit similar to the one proposed by the authors. However, to the best of our knowledge, nothing of the sort is available for multiple Markovian preferential attachment structures. This would be an interesting direction to explore.

摘要来源：OpenAlex 索引摘要（稿件版本未注明）