This paper produces an efficient Semidefinite Programming (SDP) solution for community detection that incorporates non-graph data, which in this context is known as side information. SDP is an efficient solution for standard community detection on graphs. We formulate a semi-definite relaxation for the maximum likelihood estimation of node labels, subject to observing both graph and non-graph data. This formulation is distinct from the SDP solution of standard community detection, but maintains its desirable properties. We calculate the exact recovery threshold for three types of non-graph information, which in this paper are called side information: partially revealed labels, noisy labels, as well as multiple observations (features) per node with arbitrary but finite cardinality. We find that SDP has the same exact recovery threshold in the presence of side information as maximum likelihood with side information. Thus, the methods developed herein are computationally efficient as well as asymptotically accurate for the solution of community detection in the presence of side information. Simulations show that the asymptotic results of this paper can also shed light on the performance of SDP for graphs of modest size.

中文翻译：

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带辅助信息的社区检测的半定程序

本文为社区检测提供了一种有效的半定性编程（SDP）解决方案，该解决方案结合了非图形数据，在这种情况下，它被称为辅助信息。SDP是图形上标准社区检测的有效解决方案。我们在观察图和非图数据的前提下，针对节点标签的最大似然估计制定了半确定松弛。此公式与标准社区检测的SDP解决方案不同，但保留了其所需的属性。我们计算三种非图形信息的准确恢复阈值，在本文中将其称为辅助信息：部分显示的标签，嘈杂的标签以及每个节点具有任意但有限基数的多个观测值（特征）。我们发现，在附带信息存在的情况下，SDP具有与附带信息的最大可能性相同的确切恢复阈值。因此，对于在存在辅助信息的情况下的社区检测的解决方案，本文开发的方法在计算效率和渐近精度上都是有效的。仿真表明，本文的渐近结果也可以为中等大小的图形提供SDP的性能。