Functional control of network dynamics using designed Laplacian spectra

TL;DR

This paper presents a rigorous method for designing weighted networks with specific Laplacian spectra to control complex system dynamics, enabling targeted phenomena like chimera states, pattern suppression, and localization.

Contribution

It introduces a novel weighted graph construction framework that precisely achieves any desired spectrum, expanding spectral graph theory beyond unweighted networks.

Findings

Designed spectra can induce chimera states in oscillator networks.

Spectral design can suppress pattern formation in PDE models.

Localized states can be achieved in quantum network models.

Abstract

Complex real-world phenomena across a wide range of scales, from aviation and internet traffic to signal propagation in electronic and gene regulatory circuits, can be efficiently described through dynamic network models. In many such systems, the spectrum of the underlying graph Laplacian plays a key role in controlling the matter or information flow. Spectral graph theory has traditionally prioritized unweighted networks. Here, we introduce a complementary framework, providing a mathematically rigorous weighted graph construction that exactly realizes any desired spectrum. We illustrate the broad applicability of this approach by showing how designer spectra can be used to control the dynamics of various archetypal physical systems. Specifically, we demonstrate that a strategically placed gap induces chimera states in Kuramoto-type oscillator networks, completely suppresses pattern formation in a generic Swift-Hohenberg model, and leads to persistent localization in a discrete Gross-Pitaevskii quantum network. Our approach can be generalized to design continuous band gaps through periodic extensions of finite networks.