M. Catalano, A. Fasano, M. Giordano, and G. Rebaudo’s contribution to the Discussion of ‘Root and community inference on the latent growth process of a network’ by Crane and Xu

We congratulate the authors for their methodological and theoretical contribution to the statistical literature on networks.A natural extension of the proposed PAPER model is included, with K communities growing simultaneously and where new nodes are either assigned to an existing community or elected as a new root.The employed assignment rule is of Pólya-urn type, which leads to a logarithmic growth of the number of communities (Korwar and Hollander, 1972) and is known to coincide with the predictive scheme of exchangeable sequences associated to the Dirichlet process.The probability of creating a new community is then independent of the number of past ones, which is a distinctive feature of the Dirichlet process within the class of Gibbs-type priors (De Blasi et al., 2015).An interesting direction would be to allow for more flexible predictive schemes that ensure alternative asymptotics, ranging from power-law behaviours (via the Pitman-Yor process (Pitman, 2006) or normalized generalized gamma completely random measures (Lijoi et al., 2007)) to slower than logarithmic growth (via the single-group hierarchical Dirichlet process (Camerlenghi et al, 2018)).An important theoretical aspect is the number of communities.While the authors provide an empirical investigation, future research could tackle the question of posterior consistency for the number of communities, along the lines of the existing results for stochastic block models (Geng et al., 2019) and in the growing literature in Bayesian nonparametric mixture models (Nobile, 1994;Miller and Harrison, 2013;Ascolani et al., 2023).Turning to applications, the proposed model lends itself to some natural generalizations suggested by popular epidemiological models, like SIR dynamics, where