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

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 For a random forest of K heterogeneous disjoint component trees, denote its growth parameters by (α α α, β β β) ≡ (α k , β k ) K k=1 .We define its growth process by the HAPA(α α α, β β β, K) model: For k ∈ S = [K] ≡ {1, . . ., K}, let node k be the root of the k-th component tree.For any K ≤ t ≤ n, k ∈ [K], denote the k-th component tree at time t by T k,t , and the time labeled forest at time t by F t = ∪ K k=1 T k,t .At t = K, T k,t is the set of the singleton node k.For t = K + 1, . . ., n, given F t-1 , we add a new node t and a new random edge (t, w t ) where the existing node w t ∈ T k,t-1 is chosen with probability