Optimal Estimation of the Number of Communities

TL;DR

This paper introduces StGoF, a stepwise method for estimating the number of communities in networks, which is consistent and achieves optimal phase transition under certain signal-to-noise ratio conditions.

Contribution

The paper proposes a novel stepwise goodness-of-fit approach, StGoF, that effectively estimates the number of communities even with degree heterogeneity and varying sparsity levels, overcoming analytical challenges.

Findings

StGoF provides a consistent estimate of K.

StGoF is uniformly consistent when SNR tends to infinity.

The method achieves the optimal phase transition for community detection.

Abstract

In network analysis, how to estimate the number of communities $K$ is a fundamental problem. We consider a broad setting where we allow severe degree heterogeneity and a wide range of sparsity levels, and propose Stepwise Goodness-of-Fit (StGoF) as a new approach. This is a stepwise algorithm, where for $m = 1, 2, \ldots$, we alternately use a community detection step and a goodness-of-fit (GoF) step. We adapt SCORE \cite{SCORE} for community detection, and propose a new GoF metric. We show that at step $m$, the GoF metric diverges to $\infty$ in probability for all $m < K$ and converges to $N(0,1)$ if $m = K$. This gives rise to a consistent estimate for $K$. Also, we discover the right way to define the signal-to-noise ratio (SNR) for our problem and show that consistent estimates for $K$ do not exist if $\mathrm{SNR} \goto 0$, and StGoF is uniformly consistent for $K$ if $\mathrm{SNR} \goto \infty$. Therefore, StGoF achieves the optimal phase transition. Similar stepwise methods (e.g., \cite{wang2017likelihood, ma2018determining}) are known to face analytical challenges. We overcome the challenges by using a different stepwise scheme in StGoF and by deriving sharp results that are not available before. The key to our analysis is to show that SCORE has the {\it Non-Splitting Property (NSP)}. Primarily due to a non-tractable rotation of eigenvectors dictated by the Davis-Kahan $\sin(\theta)$ theorem, the NSP is non-trivial to prove and requires new techniques we develop.