A key challenge in network science is the detection of communities, which are sets of nodes in a network that are densely connected internally but sparsely connected to the rest of the network. A fundamental result in community detection is the existence of a nontrivial threshold for community detectability on sparse graphs that are generated by the planted partition model (PPM). Below this so-called ``detectability limit'', no community-detection method can perform better than random chance. Spectral methods for community detection fail before this detectability limit because the eigenvalues corresponding to the eigenvectors that are relevant for community detection can be absorbed by the bulk of the spectrum. One can bypass the detectability problem by using special matrices, like the non-backtracking matrix, but this requires one to consider higher-dimensional matrices. In this paper, we show that the difference in graph energy between a PPM and an Erdős--Rényi (ER) network has a distinct transition at the detectability threshold even for the adjacency matrices of the underlying networks. The graph energy is based on the full spectrum of an adjacency matrix, so our result suggests that standard graph matrices still allow one to separate the parameter regions with detectable and undetectable communities.

翻译：网络科学中的一个关键挑战是社区检测，社区指的是网络中内部连接紧密但与网络其余部分连接稀疏的节点集合。社区检测的一个基本结论是：在由植入分区模型（PPM）生成的稀疏图中，存在一个非平凡的社区可检测性阈值。低于这一所谓的“可检测性极限”，任何社区检测方法的性能都无法优于随机猜测。基于谱方法的社区检测在该可检测性极限之前就会失效，因为与社区检测相关的特征向量所对应的特征值可能被谱的主体部分所吸收。通过使用特殊矩阵（如非回溯矩阵）可以绕过可检测性问题，但这需要处理更高维度的矩阵。本文证明，即使对于底层网络的邻接矩阵，PPM 与 Erdős–Rényi（ER）网络之间的图能量差异在可检测性阈值处也存在明显转变。图能量基于邻接矩阵的完整谱，因此我们的结果表明，标准图矩阵仍可用于区分社区可检测与不可检测的参数区域。