Graph Fourier Transform Enhancement through Envelope Extensions

TL;DR

This work addresses the inability of Graph Fourier Transforms to natively handle directed graphs by embedding any digraph into Cayley digraphs on the cyclic group $\mathbb{Z}_N$ via envelope extensions, enabling a well-defined GFT and a convolution framework. The authors establish a systematic process to construct admissible Cayley extensions, analyze spectral components with measures $\Delta_{\mathbf{V}}$ and $\delta_{\mathbf{V}}$, and demonstrate a practical pipeline on a real Friendship digraph to select envelope extensions that preserve structure while enabling stable, filterable graph signals. A convolution theorem is shown to hold on admissible digraphs, and envelope extensions enable targeted, spectrally faithful GFTs that can implement shift-invariant graph filters. The approach provides a principled way to tailor Fourier bases to directed graphs, balancing spectral accuracy and numerical stability for robust graph-signal processing in applications like social networks and other directed systems.

Abstract

Many real-world networks are characterized by directionality; however, the absence of an appropriate Fourier basis hinders the effective implementation of graph signal processing techniques. Inspired by discrete signal processing, where embedding a line digraph into a cycle digraph facilitates the powerful Discrete Fourier Transform for signal analysis, addressing the structural complexities of general digraphs can help overcome the limitations of the Graph Fourier Transform (GFT) and unlock its potential. The Discrete Fourier Transform (DFT) serves as a Graph Fourier Transform for both cycle graphs and Cayley digraphs on the finite cyclic groups $\mathbb{Z}_N$. We propose a systematic method to identify a class of such Cayley digraphs that can encompass a given directed graph. By embedding the directed graph into these Cayley digraphs and opting for envelope extensions that naturally support the Graph Fourier Transform, the GFT functionalities of these extensions can be harnessed for signal analysis. Among the potential envelopes, optimal performance is achieved by selecting one that meets key properties. This envelope's structure closely aligns with the characteristics of the original digraph. The Graph Fourier Transform of this envelope is reliable in terms of numerical stability, and its columns approximately form an eigenbasis for the adjacency matrix associated with the original digraph. It is shown that the envelope extensions possess a convolution product, with their GFT fulfilling the convolution theorem. Additionally, shift-invariant graph filters (systems) are described as the convolution operator, analogous to the classical case. This allows the utilization of systems for signal analysis.

Figures (15)

• Figure 1: The directed cycle graph with 5 vertices.
• Figure 3: Hamiltonian Cycle Creation
• Figure 4: The figure shows the $(\delta,\Delta)$ indices and condition numbers of $\mathop{\mathrm{\mathbf{F}}}\nolimits_w$ for weights ranging from $1$ to $0.01$ with a step size of $0.01$ when $N=16$. The minimum and maximum condition numbers are reported as $1$ and $74.98942$.
• Figure 5: Friendship digraph $\mathop{\mathrm{\mathbf{G}}}\nolimits_{\text{f}}$ with 70 vertices.
• Figure 6: Adjacency matrix associated with Friendship digraph $\mathop{\mathrm{\mathbf{G}}}\nolimits_{\text{f}}$ clustered into three clusters using spectral clustering.

Theorems & Definitions (14)

• Definition 1
• Theorem 2
• Theorem 3
• proof
• Remark 4
• Definition 5
• Example 6
• Definition 7
• Theorem 8
• proof