Localization Phenomena in Large-Scale Networked Systems: Robustness and Fragility of Dynamics

TL;DR

The paper addresses robustness fragility arising from eigenvector localization in graph Laplacians of large networks. It combines $\varepsilon$-pseudospectra and the LTI small-gain framework to relate localization to perturbation sensitivity in networked second-order dynamics, encompassing four perturbation models (edge, global node, local node, and local-reciprocal). Through analytic perturbation results and a detailed numerical example, it shows that localized eigenvectors produce disproportionately large responses to localized disturbances, with fragility persisting in the localized region as the network grows. These findings suggest localization-driven fragility is likely generic in networked oscillator and power-grid-type systems, motivating further work to identify structural causes of localization and to develop robust design principles for large-scale networks.

Abstract

We study phenomena where some eigenvectors of a graph Laplacian are largely confined in small subsets of the graph. These localization phenomena are similar to those generally termed Anderson Localization in the Physics literature, and are related to the complexity of the structure of large graphs in still unexplored ways. Using spectral perturbation theory and pseudo-spectrum analysis, we explain how the presence of localized eigenvectors gives rise to fragilities (low robustness margins) to unmodeled node or link dynamics. Our analysis is demonstrated by examples of networks with relatively low complexity, but with features that appear to induce eigenvector localization. The implications of this newly-discovered fragility phenomenon are briefly discussed.

Figures (5)

• Figure 1: An example illustrating Laplacian eigenvector localization. A subset of the eigenvectors are localized, while others are not. The basic "banded" Laplacian structure is shown in (a). Different network sizes $N$ are used in the various subplots for ease of visualization.
• Figure 2: The setting of the robust stability Small-Gain Theorem \ref{['small_gain.thm']}. An uncertain system is represented as the feedback interconnection between the known dynamics $M$ and the unknown perturbation $\textcolor{dred}{\bm{\Delta}}$, which is itself a dynamical system. Robust stability is the question of whether the perturbed system above is stable for all possible perturbations $\textcolor{dred}{\bm{\Delta}}$ in the specified class (e.g. norm bounded).
• Figure 3: The contrast in robustness between node perturbations (\ref{['glob_node_pert.eq']}) of a localized versus a delocalized node in the network of Figure \ref{['TM_p_spec_1.fig']} with $N=200$. In this case, the Laplacian's spectrum is about an order of magnitude more sensitive to localized node perturbation in comparison to perturbing nodes in the delocalized region.
• Figure 4: Node perturbations in the localized (red) and delocalized (blue) regions for the uncertain 2nd-order oscillator model (\ref{['Eq:sys']}). $\epsilon$-pseudospectra (in green) of the overall $A$-matrix \ref{['A_mat_big.eq']} for the same value of $\varepsilon$ are shown for each case. The eigenvalues themselves are color coded with red and blue for localized and delocalized eigenvalues respectively. We zoom in on the eigenvalues with a positive imaginary part; the rest of the eigenvalues are a mirror image (complex conjugates) of the ones shown.
• Figure 5: Node, edge and eigenvalue sensitivities for the Laplacian of the example of Figure \ref{['TM_p_spec_1.fig']} with $N=1000$ and band size $\approx 500$. The conclusions are consistent with the results shown in Figure \ref{['TM_p_spec.fig']}.

Theorems & Definitions (4)

• Theorem II.1
• Definition II.2
• Theorem II.3
• Definition V.1