Density Evolution in the Degree-correlated Stochastic Block Model
This work provides a refined theoretical analysis for community detection in networks, offering exact misclassification bounds for a specific model, which is incremental to existing threshold recovery studies.
The paper tackles the problem of determining the average fraction of misclassified vertices in the degree-correlated stochastic block model, showing that the minimum misclassification fraction is given by Q(√v*) and is achieved by belief propagation in linear time.
There is a recent surge of interest in identifying the sharp recovery thresholds for cluster recovery under the stochastic block model. In this paper, we address the more refined question of how many vertices that will be misclassified on average. We consider the binary form of the stochastic block model, where $n$ vertices are partitioned into two clusters with edge probability $a/n$ within the first cluster, $c/n$ within the second cluster, and $b/n$ across clusters. Suppose that as $n \to \infty$, $a= b+ μ\sqrt{ b} $, $c=b+ ν\sqrt{ b} $ for two fixed constants $μ, ν$, and $b \to \infty$ with $b=n^{o(1)}$. When the cluster sizes are balanced and $μ\neq ν$, we show that the minimum fraction of misclassified vertices on average is given by $Q(\sqrt{v^*})$, where $Q(x)$ is the Q-function for standard normal, $v^*$ is the unique fixed point of $v= \frac{(μ-ν)^2}{16} + \frac{ (μ+ν)^2 }{16} \mathbb{E}[ \tanh(v+ \sqrt{v} Z)],$ and $Z$ is standard normal. Moreover, the minimum misclassified fraction on average is attained by a local algorithm, namely belief propagation, in time linear in the number of edges. Our proof techniques are based on connecting the cluster recovery problem to tree reconstruction problems, and analyzing the density evolution of belief propagation on trees with Gaussian approximations.