Rates of Convergence of Spectral Methods for Graphon Estimation

This paper studies the problem of estimating the graphon function – a generative mechanism for a class of random graphs that are useful approximations to real networks. Specifically, a graph of $n$ vertices is generated such that each pair of two vertices $i$ and $j$ are connected independently with probability $\rho_n \times f(x_i,x_j)$, where $x_i$ is the unknown $d$-dimensional label of vertex $i$, $f$ is an unknown symmetric function, and $\rho_n$, assumed to be $\Omega(\log n/n)$, is a scaling parameter characterizing the graph sparsity. The task is to estimate graphon $f$ given the graph. Recent studies have identified the minimax optimal estimation error rate for $d=1$. However, there exists a wide gap between the known error rates of polynomial-time estimators and the minimax optimal error rate. We improve on the previously known error rates of polynomial-time estimators, by analyzing a spectral method, namely universal singular value thresholding (USVT) algorithm. When $f$ belongs to either Hölder or Sobolev space with smoothness index $\alpha$, we show the error rates of USVT are at most $(n\rho)^{ -2 \alpha / (2\alpha+d)}$. These error rates approach the minimax optimal error rate $\log (n\rho)/(n\rho)$ proved in prior work for $d=1$, as $\alpha$ increases, i.e., $f$ becomes smoother. Furthermore, when $f$ is analytic with infinitely many times differentiability, we show the error rate of USVT is at most $\log^d (n\rho)/(n\rho)$. When $f$ is a step function which corresponds to the stochastic block model with $k$ blocks for some $k$, the error rate of USVT is at most $k/(n\rho)$, which is larger than the minimax optimal error rate by at most a multiplicative factor $k/\log k$. This coincides with the computational gap observed in community detection. A key ingredient of our analysis is to derive the eigenvalue decaying rate of the edge probability matrix using piecewise polynomial approximations of the graphon function $f$.