The stochastic block model is a canonical random graph model for clustering and community detection on network-structured data. Decades of extensive study on the problem have established many profound results, among which the phase transition at the Kesten-Stigum threshold is particularly interesting both from a mathematical and an applied standpoint. It states that no estimator based on the network topology can perform substantially better than chance on sparse graphs if the model parameter is below a certain threshold. Nevertheless, if we slightly extend the horizon to the ubiquitous semi-supervised setting, such a fundamental limitation will disappear completely. We prove that with an arbitrary fraction of the labels revealed, the detection problem is feasible throughout the parameter domain. Moreover, we introduce two efficient algorithms, one combinatorial and one based on optimization, to integrate label information with graph structures. Our work brings a new perspective to the stochastic model of networks and semidefinite program research.

翻译：随机块模型是处理网络结构数据聚类和社区检测的经典随机图模型。对该问题长达数十年的深入研究已确立了许多深刻结论，其中Kesten-Stigum阈值处的相变现象无论从数学还是应用角度都特别引人注目。该阈值表明：当模型参数低于该阈值时，基于网络拓扑结构的任何估计器在稀疏图上的表现都无法显著优于随机猜测。然而，若将研究视野稍作扩展至普遍存在的半监督场景，这一根本性限制将完全消失。我们证明，只要任意比例的标签被揭示，整个参数域内的检测问题都是可行的。此外，我们提出了两种高效算法——一种基于组合方法，另一种基于优化方法——以将标签信息与图结构相融合。我们的工作为网络随机模型及半定规划研究带来了全新视角。