Inferences on Parameters in Severely Heterogeneous Degree Corrected Stochastic Block Models

Abstract

With the rise of big data, networks have pervaded many aspects of our daily lives, with applications ranging from the social to natural sciences. Understanding the latent structure of network is thus an important question. In this paper, we model the network using a Degree-Corrected Mixed Membership (DCMM) model, in which every node $i$ has an intrinsic membership vector $\pi_i = (\pi_1, \cdots \pi_K)$ measuring its belonging to one of $K$ communities. Our central aim is to construct inferential procedures for the probability matrix and degree parameters $P$ and $\Theta$ of the DCMM, an often overlooked question in the literature. By providing new procedures, we empower practicioners to answer various ranking and dynamics questions related to networks. These questions may prove to be impactful, as they may aid in identifying non-tradiational canditates for targeted therapies and detecting subtle shifts within network time series, among other applications. At the end of our work, for example, we present an application for detecting changepoints in real-world global trade networks, revealing a significant changepoint that concides with the year corresponding to the global financial crisis.