Zero-padding on Connected Directed Acyclic Graphs for Spectral Processing

Directed acyclic graphs (DAGs) are often utilized 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. To address this limitation, we propose, in this paper, a zero-padding approach for connected DAGs. This approach involves augmenting the original connected DAG with additional nodes that are connected to the existing structure. These added nodes are characterized by signal values set to zero and does not interfere with the original DAG except at the source and sink. The proposed technique enables the spectral evaluation of system outputs on DAGs, that is the computation of vertex-domain convolution (output of a graph filter) in without the adverse effects of aliasing due to the added nodes (added nodes do not change the original output of a filter on a DAG).