This paper proposes a Generalized Power Method (GPM) to tackle the problem of community detection and group synchronization simultaneously in a direct non-convex manner. Under the stochastic group block model (SGBM), theoretical analysis indicates that the algorithm is able to exactly recover the ground truth in $O(n\log^2n)$ time, sharply outperforming the benchmark method of semidefinite programming (SDP) in $O(n^{3.5})$ time. Moreover, a lower bound of parameters is given as a necessary condition for exact recovery of GPM. The new bound breaches the information-theoretic threshold for pure community detection under the stochastic block model (SBM), thus demonstrating the superiority of our simultaneous optimization algorithm over the trivial two-stage method which performs the two tasks in succession. We also conduct numerical experiments on GPM and SDP to evidence and complement our theoretical analysis.

翻译：本文建议采用通用功率法(GPM)解决社区探测和群体同步问题,以直接非凝固的方式同时解决社区探测和群体同步问题。在随机群块模型(SGBM)下,理论分析表明算法能够用美元(n\log=2n)时间准确恢复地面真相,大大优于半确定性编程基准方法(SDP)用美元(n=3.5})时间。此外,设定较低的参数范围是精确恢复GPM的必要条件。新的约束违反了根据随机群块模型(SBM)进行纯社区探测的信息理论阈值,从而表明我们同时使用的优化算法优于连续执行两项任务的微小的两阶段方法。我们还对GPM和SDP进行数字实验以作为证据并补充我们的理论分析。