Eigenvector localization in Networked Dynamical Systems

Poorva Shukla

Advisor: Bassam Bamieh

Friday October 2nd, 12 pm

Henley 1010

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https://ucsb.zoom.us/j/3260278731

Meeting ID: 326 027 8731 

 

Abstract: Eigenvectors are said to be localized when most of their mass is concentrated on a small region of the underlying domain. Localization is classically associated with Anderson localization in disordered media, but it also arises in deterministic systems with complex geometry. This dissertation studies eigenvector localization for graph Laplacians and related operators that govern networked dynamical systems, such as second-order oscillator networks, power grids, vehicular formations, and consensus protocols. It addresses three questions: what localization implies for network dynamics, what causes it, and how its different sources can be analyzed within a single framework.

We first examine the dynamical consequences of localization. Using spectral perturbation theory and pseudospectral analysis, we show that localized eigenvectors create fragilities. Unmodeled node or edge dynamics in localized regions of a network reduce its stability margins far more than the same perturbations in delocalized regions, and this contrast can grow without bound as the network grows. We then show that classical graph metrics, such as node degree, graph distance, and resistance distance, fail to predict how an impulse spreads through a heterogeneous oscillator network. We introduce modal distance, a measure of the overlap between the modal coordinates of two nodes, which explains two phenomena: node-dependent propagation and nonlocal excitation. Using a feedback formulation, we further show that the eigenvector destabilized by a local dynamical perturbation is localized regardless of where the perturbation enters the system.

We next study the causes of localization. We prove that if a matrix has entries that decay away from a sparse set and has an isolated eigenvalue, then the corresponding eigenvector is localized and decays at the same rate as the entries of the matrix inverse. We establish this for sparse, exponentially decaying, and polynomially decaying entries. We then examine the structural origins of localization from two perspectives. From a spectral perspective, we consider Laplacians of the form $\cL = \bar\cL + E$ with nearly orthogonal eigenbases and relative spectral concentration. For such Laplacians, we observe that the spectrum of $\cL$ is close to a uniform shift of the spectrum of $E$ and that the eigenvectors of $\cL$ are similar in shape to those of $E$. We introduce a notion of shape similarity that is weaker than closeness in angle to capture this. From a geometric perspective, we demonstrate that, in both continuum and discrete settings, eigenfunctions localize where the Ricci curvature varies.

Finally, we develop unifying frameworks. We compare the empirical spectral distributions arising in geometric and Anderson localization by expressing both as conjugation by structured unitary matrices. We adapt localization landscape theory to graph Laplacians through a Neumann landscape function, given by the diagonal of the Laplacian pseudoinverse. This function predicts localization regions without computing eigenvectors and yields bounds on eigenvector decay. We close with a feedback framework that treats geometric localization, Anderson localization, and localization due to local dynamical perturbations within a single analysis.