Optimal Estimation of Parameters in Degree Corrected Mixed Membership 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 the 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 affinity parameter $\theta_i$, measuring the degree of connectivity, and an intrinsic membership probability vector $\pi_i = (\pi_1, \cdots \pi_K)$, measuring its belonging to one of $K$ communities, and a probability matrix $P$ that describes the average connectivity between two communities. Our central question is to determine the optimal estimation rates for the probability matrix and degree parameters $P$ and $\Theta$ of the DCMM, an often overlooked question in the literature. By providing new lower bounds, we show that simple extensions of existing estimators in the literature indeed achieve the optimal rate. Simulations lend further support to our theoretical results.