A fundamental problem in mathematics and network analysis is to find conditions under which a graph can be partitioned into smaller pieces. The most important tool for this partitioning is the Fiedler vector or discrete Cheeger inequality. These results relate the graph spectrum (eigenvalues of the normalized adjacency matrix) to the ability to break a graph into two pieces, with few edge deletions. An entire subfield of mathematics, called spectral graph theory, has emerged from these results. Yet these results do not say anything about the rich community structure exhibited by real-world networks, which typically have a significant fraction of edges contained in numerous densely clustered blocks. Inspired by the properties of real-world networks, we discover a new spectral condition that relates eigenvalue powers to a network decomposition into densely clustered blocks. We call this the \emph{spectral triadic decomposition}. Our relationship exactly predicts the existence of community structure, as commonly seen in real networked data. Our proof provides an efficient algorithm to produce the spectral triadic decomposition. We observe on numerous social, coauthorship, and citation network datasets that these decompositions have significant correlation with semantically meaningful communities.

翻译：数学与网络分析中的一个基本问题是寻找将图划分为更小子图的条件。实现这一划分的最重要工具是Fiedler向量或离散Cheeger不等式。这些结果将图的谱（归一化邻接矩阵的特征值）与将图划分为两个部分（仅需删除少量边）的能力相关联。由此衍生了数学的一个完整子领域——谱图理论。然而，这些结果并未揭示真实世界网络所呈现的丰富社区结构，这类网络通常具有显著比例边包含于大量密集聚类块中的特征。受真实世界网络特性的启发，我们发现了新的谱条件，该条件将特征值幂次与网络分解为密集聚类块的能力相关联。我们将此称为"谱三元组分解"。该关系式精确预测了真实网络数据中常见的社区结构的存在性。我们的证明提供了生成谱三元组分解的高效算法。通过对多个社交网络、合著网络和引文网络数据集的观察，我们发现这些分解与具有语义意义的社区存在显著相关性。