The high-level structure of a graph is a crucial ingredient for the analysis and visualization of relational data. However, discovering the salient graph patterns that form this structure is notoriously difficult for two reasons. (1) Finding important patterns, such as cliques and bicliques, is computationally hard. (2) Real-world graphs contain noise, and therefore do not always exhibit patterns in their pure form. Defining meaningful noisy patterns and detecting them efficiently is a currently unsolved challenge. In this paper, we propose to use well-ordered matrices as a tool to both define and effectively detect noisy patterns. Specifically, we represent a graph as its adjacency matrix and optimally order it using Moran's $I$. Standard graph patterns (cliques, bicliques, and stars) now translate to rectangular submatrices. Using Moran's $I$, we define a permitted level of noise for such patterns. A combination of exact algorithms and heuristics allows us to efficiently decompose the matrix into noisy patterns. We also introduce a novel motif simplification that visualizes noisy patterns while explicitly encoding the level of noise. We showcase our techniques on several real-world data sets.

翻译：图的高层结构是关系数据分析和可视化的关键要素。然而，发现构成这种结构的显著图模式存在两大公认难点：（1）寻找重要模式（如团和双团）在计算上具有难度；（2）真实世界图数据包含噪声，因此模式往往不以纯净形式呈现。如何定义有意义的带噪模式并高效检测，仍是当前未解决的挑战。本文提出以有序矩阵作为工具，既能定义又能有效检测带噪模式。具体而言，我们将图表示为邻接矩阵，并利用Moran's $I$对其进行最优排序。标准图模式（团、双团、星形结构）在此框架下转化为矩形子矩阵。基于Moran's $I$，我们为这类模式定义了容许噪声水平。通过精确算法与启发式算法的结合，我们能够将矩阵高效分解为带噪模式。此外，我们引入一种新颖的模体简化方法，可在显式编码噪声水平的同时可视化带噪模式。我们在多个真实数据集上展示了所提技术的有效性。