现实世界网络的谱三元分解
Spectral Triadic Decompositions of Real-World Networks
AI总结：
本文提出谱三元分解这一新谱条件，将特征值幂与网络密集块分解联系起来，并给出高效算法，在社交、合著和引文网络数据上验证了其与语义社区的高度相关性。
摘要：
数学和网络分析中的一个基本问题是寻找图可以被划分为更小部分的条件。用于这种划分的最重要工具是Fiedler向量或离散Cheeger不等式。这些结果将图谱（归一化邻接矩阵的特征值）与将图分解为两部分且仅需少量边删除的能力联系起来。一个完整的数学子领域，称为谱图理论，已经从这些结果中发展出来。然而，这些结果并没有说明现实世界网络所表现出的丰富社区结构，这些网络通常有相当大比例的边包含在众多密集聚集的块中。受现实世界网络性质的启发，我们发现了一个新的谱条件，它将特征值幂与网络分解为密集聚集块联系起来。我们称之为谱三元分解。我们的关系精确预测了社区结构的存在，正如在现实网络数据中常见的那样。我们的证明提供了一种高效算法来生成谱三元分解。我们在众多社交、合著和引文网络数据集上观察到，这些分解与语义上有意义的社区具有显著相关性。
英文摘要：
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.
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