Wang, Yu and Rinaldo’s contribution to the Discussion of ‘Root and community inference on the latent growth process of a network’ by Crane and Xu

We congratulate Prof. Crane and Dr Xu for introducing a very interesting and appealing statistical model for Markovian network growth.The PAPER model enjoys several qualities.In particular, (i) it strikes a rare balance between analytic elegance and tractability, and expressive power; (ii) it allows for practicable algorithms for statistical inference that scale to networks of very large size and (iii) it is very flexible and can be readily generalized to model a variety of phenomena and features commonly observed in modern networks.As pointed out by the authors, there are many extensions and open problems related to the model that are worth considering.In this note, we suggest possible extensions to change point analysis (CPA) for networks.CPA is a well-studied topic concerned with modelling and detecting abrupt changes in the data-generating distribution in time series data.Developing models, theories and methods for CPA in dynamic and large networks is a relatively new and exciting area of research (e.g.Wang et al., 2021; Yu et al., 2023).We believe the PAPER model provides an excellent reference framework for building powerful and realistic Markovian network CP models.To provide some details, in the PAPER model, at time point t ∈ [n], the newly added node with label t is connected to an existing node w t ∈ [t -1], with probability {βD T t-1 (w t ) + α}/{β2(t -2) + α(t -1)}.The parameters (α, β) are fixed across the whole time course.It would be interesting to consider the scenario where the parameters are instead allowed to change in a piecewise manner at unknown CPs.In the simplest instantiation of the PAPER CP model, there might exist an unknown CP t * such that the values of the parameters α and β change after t * .A possible application could be, in a social network, at the booming stage of a key opinion leader, β, the parameter characterizing the 'rich gets richer' phenomenon, is positive and large.As the craze cools down, β should decrease to reflect the fading of fame.In extreme cases, β may even change the sign.To estimate the change time t * , one could consider a likelihood-based 0 -penalization (e.g.Wang et al., 2023), with the likelihood stemming from the PAPER model.A second, more sophisticated extension is to consider an APA model with multiple root nodes; at the CP(s), the number of root nodes, along, possibly, with the model parameter, change, thus accounting for the creation, elimination or even merging of tree components.Finally, the CPA tasks just outlined can be analyzed in the offline settings, in which the fully grown network at a given time, say n, is observed and then analyzed.Lastly, we congratulate Prof. Crane and Dr Xu again for their excellent paper.