A Spectral Graph Uncertainty Principle

The spectral theory of graphs provides a bridge between classical signal processing and the nascent field of graph signal processing. In this paper, a spectral graph analogy to Heisenberg's celebrated uncertainty principle is developed. Just as the classical result provides a tradeoff between signal localization in time and frequency, this result provides a fundamental tradeoff between a signal's localization on a graph and in its spectral domain. Using the eigenvectors of the graph Laplacian as a surrogate Fourier basis, quantitative definitions of graph and spectral "spreads" are given, and a complete characterization of the feasibility region of these two quantities is developed. In particular, the lower boundary of the region, referred to as the uncertainty curve, is shown to be achieved by eigenvectors associated with the smallest eigenvalues of an affine family of matrices. The convexity of the uncertainty curve allows it to be found to within $\varepsilon$ by a fast approximation algorithm requiring $O(\varepsilon^{-1/2})$ typically sparse eigenvalue evaluations. Closed-form expressions for the uncertainty curves for some special classes of graphs are derived, and an accurate analytical approximation for the expected uncertainty curve of Erd\H{o}s-R\'enyi random graphs is developed. These theoretical results are validated by numerical experiments, which also reveal an intriguing connection between diffusion processes on graphs and the uncertainty bounds.Comment: 40 pages, 8 figure

On unique continuation for solutions of the Schr{\"o}dinger equation on trees

We prove that if a solution of the time-dependent Schr{\"o}dinger equation on an homogeneous tree with bounded potential decays fast at two distinct times then the solution is trivial. For the free Schr{\"o}dinger operator, we use the spectral theory of the Laplacian and complex analysis and obtain a characterization of the initial conditions that lead to a sharp decay at any time. We then use the recent spectral decomposition of the Schr{\"o}dinger operator with compactly supported potential due to Colin de Verdi{\`e}rre and Turc to extend our results in the presence of such potentials. Finally, we use real variable methods first introduced by Escauriaza, Kenig, Ponce and Vega to establish a general sharp result in the case of bounded potentials

Extending Modular Semantics for Bipolar Weighted Argumentation (Technical Report)

Weighted bipolar argumentation frameworks offer a tool for decision support and social media analysis. Arguments are evaluated by an iterative procedure that takes initial weights and attack and support relations into account. Until recently, convergence of these iterative procedures was not very well understood in cyclic graphs. Mossakowski and Neuhaus recently introduced a unification of different approaches and proved first convergence and divergence results. We build up on this work, simplify and generalize convergence results and complement them with runtime guarantees. As it turns out, there is a tradeoff between semantics' convergence guarantees and their ability to move strength values away from the initial weights. We demonstrate that divergence problems can be avoided without this tradeoff by continuizing semantics. Semantically, we extend the framework with a Duality property that assures a symmetric impact of attack and support relations. We also present a Java implementation of modular semantics and explain the practical usefulness of the theoretical ideas