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

The $\beta$-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 $\beta$-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 $\max_t \theta_t \to 0$ and $\theta_t \|\theta\|_1 \to \infty$, 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 $\hat{\beta}_t$ up to a constant factor. Here, $\hat{\beta}_t$ is the CCR estimator of the node-specific parameter $\beta_t$, $\theta_t = \exp(\beta_t)$ and $\theta=(\theta_1, \ldots, \theta_n)$. Even if the whole network density is close to the Erd\H{o}s-R\'{e}nyi lower bound $\log n/n$, the CCR estimator for the single parameter $\beta_t$ is still consistent as long as $\theta_t \|\theta\|_1 \to \infty$. 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 $\theta_t \|\theta\|_1$. Numerical studies and an application to a sparse network data set demonstrate our theoretical findings.