Directed acyclic graphs (DAGs) are used for modeling causal relationships, dependencies, and flows in various systems. However, spectral analysis becomes impractical in this setting because the eigendecomposition of the adjacency matrix yields all eigenvalues equal to zero. This inherent property of DAGs results in an inability to differentiate between frequency components of signals on such graphs. This problem can be addressed by adding edges in DAG. However, this approach changes the physics of the considered problem. To address this limitation, we propose a graph zero-padding approach. This approach involves augmenting the original DAG with additional vertices that are connected to the existing structure. The added vertices are characterized by signal values set to zero. The proposed technique enables the spectral evaluation of system outputs on DAGs, that is the computation of vertex-domain convolution without the adverse effects of aliasing due to changes in graph structure.

翻译：有向无环图（DAG）用于建模各类系统中的因果关系、依赖关系及信息流。然而，由于邻接矩阵的特征分解会产生全零特征值，在此场景下频谱分析变得不可行。DAG的固有特性导致无法区分此类图上信号的频率分量。现有方法通过在DAG中添加边来解决该问题，但这改变了所研究问题的物理本质。为突破这一局限，我们提出图零填充方法。该方法通过在原DAG中增加与现有结构相连的额外顶点来实现增强，这些新增顶点的信号值被设为零。所提技术使得DAG上系统输出的频谱评价成为可能，即能够在避免因图结构变化导致混叠效应的前提下完成顶点域卷积计算。