Fourier Analysis of Signals on Directed Acyclic Graphs (DAG) Using Graph Zero-Padding

TL;DR

The proposed technique enables the spectral evaluation of system outputs on DAGs without the adverse effects of aliasing due to changes in a graph structure, with the ultimate goal of preserving the output of the system on a graph as if the changes in the graph structure were not performed.

Abstract

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 {alternating the Fourier basis or adding edges in a DAG}. However, these approaches change the physics of the considered problem. To address this limitation, we propose a \textit{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 (in almost all cases), that is the computation of vertex-domain convolution without the adverse effects of aliasing due to changes in a graph structure, { with the ultimate goal of preserving the output of the system on a graph as if the changes in the graph structure were not performed}.

Figures (13)

• Figure 1: Examples of a directed acyclic graph (DAG): (a) Disconnected DAG (for example, a path connecting vertices 2 and 3 does not exist), (b) Connected DAG (there exists a path between any pair of vertices). A Hamiltonian path $1\to2\to3\to4\to5\to6\to 7\to 8$, which visits all vertices of the connected DAG exactly once, is denoted by black lines.
• Figure 2: (a) The domain of classical discrete-time signal of length $N=8$. (b) The domain of classical signal commonly used in DFT calculations with $N=8$ (for example, as in seifert2021digraph). (c) The domain of the zero-padded classical signal required for DFT-based calculations of the output of an FIR system order $M\leq 3$, with $N=8$.
• Figure 3: Signal $x(n)$ of length $N=8$ on a graph as its domain (first row), along with shifted versions of the same signal (second and third rows). The last sub-figure represents the output of a system on a graph of order $S=2$ with system coefficients equal to $1/3$.
• Figure 4: (a) Signal $x(n)$ of length $N=8$ on a graph as its domain. (b) Signal on a graph as it must be used in the DFT output calculation on a FIR system whose order is $S \le 3$. (c) Signal $x(n-1)$ shifted on a graph from (b). (d) Signal $x(n-2)$ on a graph from (b). (e) Output signal as a circular (DFT) convolution $\mathbf{y}=\mathbf{x}*[\frac{1}{3},\frac{1}{3},\frac{1}{3}]$ equal (within the basic period) to the aperiodic convolution of the same signal and system.
• Figure 5: An illustration of a zero-padded DAG. The DAG shown in Fig. \ref{['DAG_E']} (b) is zero-padded with additional vertices (marked in red) and edges (highlighted in green), necessary for the GFT-based analysis.