We study community detection in the contextual stochastic block model arXiv:1807.09596 [cs.SI], arXiv:1607.02675 [stat.ME]. In arXiv:1807.09596 [cs.SI], the second author studied this problem in the setting of sparse graphs with high-dimensional node-covariates. Using the non-rigorous cavity method from statistical physics, they conjectured the sharp limits for community detection in this setting. Further, the information theoretic threshold was verified, assuming that the average degree of the observed graph is large. It is expected that the conjecture holds as soon as the average degree exceeds one, so that the graph has a giant component. We establish this conjecture, and characterize the sharp threshold for detection and weak recovery.

中文翻译：

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上下文随机块模型：尖锐的阈值和连续性

我们在上下文随机块模型 arXiv:1807.09596 [cs.SI]、arXiv:1607.02675 [stat.ME] 中研究社区检测。在 arXiv:1807.09596 [cs.SI] 中，第二作者在具有高维节点协变量的稀疏图的设置中研究了这个问题。使用来自统计物理学的非严格空腔方法，他们推测了在这种情况下社区检测的严格限制。进一步验证了信息论阈值，假设观察到的图的平均度数很大。预计只要平均度数超过 1，该猜想就成立，因此该图具有巨大的分量。我们建立了这个猜想，并表征了检测和弱恢复的尖锐阈值。