Refining Directed Graphs for Spectral Analysis through Strategic Topological Modifications

In graph signal processing, the absence of a well-defined graph Fourier transform complicates spectral analysis on directed graphs. To address this, we propose a technique that subtly modifies the graph’s topology, making minimal changes to facilitate spectral analysis.Our approach starts by adding edges to eliminate sources and sinks, followed by identifying connections that impede the non-singularity and diagonalizability of the adjacency matrix. We then make minor edge weight adjustments to achieve diagonalizability, ensuring efficient spectral analysis of graph signals defined on directed graph.