When Slepian Meets Fiedler: Putting a Focus on the Graph Spectrum
This work addresses the need for localized frequency analysis in graph signal processing, which is incremental as it builds on existing graph spectral methods.
The paper tackles the problem of designing graph signals with concentrated energy in specific subgraphs by introducing Slepian graph signals, establishing a novel link with Laplacian embedding and graph clustering to provide meaning to localized graph frequencies.
The study of complex systems benefits from graph models and their analysis. In particular, the eigendecomposition of the graph Laplacian lets emerge properties of global organization from local interactions; e.g., the Fiedler vector has the smallest non-zero eigenvalue and plays a key role for graph clustering. Graph signal processing focusses on the analysis of signals that are attributed to the graph nodes. The eigendecomposition of the graph Laplacian allows to define the graph Fourier transform and extend conventional signal-processing operations to graphs. Here, we introduce the design of Slepian graph signals, by maximizing energy concentration in a predefined subgraph for a graph spectral bandlimit. We establish a novel link with classical Laplacian embedding and graph clustering, which provides a meaning to localized graph frequencies.