Subgraph counting estimation for the $β$-model in sparse networks

Abstract

The $β$-model is popular for characterizing the commonly observed degree heterogeneity phenomenon in real-world networks. In this study, we develop a cycle counting approach to estimate $n$ node-specific parameters in the $β$-model for moderate or extremely sparse networks. Our proposed estimators, called \emph{Cycle Counting Ratio (CCR) Estimator}, are based on the log-ratios of two network cycle counting statistics with explicit expressions and therefore easy to compute. We focus on conditions to guarantee statistical properties of the single estimator for each node. Under the very weak conditions that $\max_t θ_t \to 0$ and $θ_t \|θ\|_1 \to \infty$, we show that the CCR estimator is consistent and achieves the minimax rate in terms of the mean squared error, which is the squared signal-to-noise ratio for $\hatβ_t$ up to a constant factor. Here, $\hatβ_t$ is the CCR estimator of the node-specific parameter $β_t$, $θ_t = \exp(β_t)$ and $θ=(θ_1, \ldots, θ_n)$. Even if the whole network density is close to the Erdős-Rényi lower bound $\log n/n$, the CCR estimator for the single parameter $β_t$ is still consistent as long as $θ_t \|θ\|_1 \to \infty$. To the best of our knowledge, this is the first time to derive the minimax rate and consistency result under such weak conditions. Under a slight stronger condition, we further establish its uniform consistency and asymptotic normality, whose asymptotic variance is $θ_t \|θ\|_1$. Numerical studies and an application to a sparse network data set demonstrate our theoretical findings.