Yang Feng and Jiajin Sun’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 Drs. Crane and Xu for their fascinating piece of work. In the multiple roots PAPER model, one assumption is that different trees share the same affine preferential attachment (APA) growth parameters (α,β)⁠. In practice, the multiple trees in a network may have varying growth mechanisms, and it may be of interest to model each tree to have its own growth parameters. We consider such a heterogeneous affine preferential attachment (HAPA) model. Definition 1 resembles the K roots model in Crane Xu (2023), with two major differences: (i) instead of homogeneous (α,β) parameters, the new model has K pairs of growth parameters, one for each tree and (ii) the likelihood of the HAPA forest depends on not only the degree sequence but also the tree growth history. EM algorithm of estimation of (α,β)=(αk,βk)k=1K under the HPAPER model EM algorithm of estimation of (α,β)=(αk,βk)k=1K under the HPAPER model Gibbs sampling of ordering and forests from the HPAPER model Gibbs sampling of ordering and forests from the HPAPER model To conduct inference for the model in Definition 1, we need to address two primary tasks: estimating the (αk,βk)k=1K parameters, and sampling from the distribution of time labelled forests. Both tasks bring new challenges compared with the K roots model in Crane and Xu (2023). First, the growth parameters can be estimated in the PAPER model without knowing the class assignments of the nodes, while in the HPAPER model, one must have some knowledge or estimation of the clustering as they are necessary even for defining the growth parameters. Second, some of the smart sampling techniques for the APA model are built upon the property that the forest likelihood depends only on the degree sequences of its nodes, which is no longer the case for the HAPA likelihood (1). Following the notation in Crane and Xu (2023), we write f~n=πfn where f~n denotes the randomly alphabetically labelled forest and π denotes the ordering of the nodes. We describe an Expectation–Maximization (EM) algorithm framework to estimate the parameters (α,β) and a Gibbs sampling framework to sample forest f~n and ordering π in Algorithms 1 and 2, respectively. We plan to give a full treatment to the proposed models and algorithms in a follow-up work. In the future, it is also of interest to establish an ‘HPAPER-SBM’ model, in which the Erdős–Rényi parameter is allowed to be different within and between different communities, and further explore suitable sampling and estimation procedures under that model. Another interesting direction is to incorporate nodal or edge covariates to the model (Huang et al., 2023; Weng & Feng, 2022).

摘要来源：OpenAlex 索引摘要（稿件版本未注明）