Vertex Censored Stochastic Kronecker Product Graphs

Stochastic Kronecker Product Graphs are an interesting and useful class of Generative Network Models. They can be fitted using a fast Maximum Likelihood Estimator and reproduce many important statistical properties commonly found in large real-world networks. However, they suffer from an unfortunate drawback: the need to pad the Stochastic Kronecker Product Graph with isolated vertices. To address this issue, we propose the use of a Vertex Censored Stochastic Kronecker Product Graph model. Using a Sequential Importance Sampling scheme, we demonstrate that this class of models can be fit in similar amount of computation time and give significant improvements in fit as measured by Maximum Likelihood. An interesting family of generative models for graphs is Stochastic Kronecker Product Graphs[8]. What makes this interesting is that Kronecker graphs seem to be appropriate for modeling many real world networks, as they can mimic several properties observed to be prevalent. For example, they are shown to have multinomial degree distribution, which with properly chosen parameters can produce heavy-tails. Give each vertex in the initiator graph self-loops, and Kronecker graphs can be made to have (small) constant diameter. Eigenvalues can be shown to be related to the degree distribution, and can be made heavy tailed. As noted in [5], Kronecker Product Graphs typically encapsulate coreperipherery type networks due to the recursive nature of the Kronecker Product Graph formulation. Many social, informational, and biological networks are known to have dense cores, making Stochastic Kronecker Product Graph networks appropriate for modeling. And perhaps most importantly, Stochastic Kronecker Product Graphs can be fit using a linear (O(E)) time algorithm via a Maximum Likelihood Estimation algorithm called KronFit developed by Leskovec et al [6]. As defined in [6], Kronecker Product Graphs start with an initiator matrix Kk ∈ RN1×N1 , we then form the Kronecker Product Graph adjacency matrix Kk = ⊗K1 (1) where ⊗ is the matrix Kronecker product. See [5] for definitions and many interesting properties. As a consequence of this construction, the Kronecker