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.
Highlights
INTRODUCTIONSpectral band gaps control the behavior of physical systems in areas as diverse as topological insulators [1,2], phononic crystals [3], superconductors [4], acoustic metamaterials [5], and active matter [6]
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
Spectral band gaps control the behavior of physical systems in areas as diverse as topological insulators [1,2], phononic crystals [3], superconductors [4], acoustic metamaterials [5], and active matter [6]
Summary
Spectral band gaps control the behavior of physical systems in areas as diverse as topological insulators [1,2], phononic crystals [3], superconductors [4], acoustic metamaterials [5], and active matter [6]. In addition to ubiquitous physical network models [7,8,9,10] ranging from aviation [11] to electronics [12], there is considerable interest in virtual or computational networks [13] with fewer physical constraints, such as those recently used to create spiral-wave chimeras in coupled chemical oscillators [14] Often, dynamics in such systems depend on the graph Laplacian [15,16] and in particular on its spectrum of eigenvalues. Our construction of networks with specified eigenvalues allows us to place arbitrary gaps in the spectrum of the network Laplacian L 1⁄4 D − A, where D and A are the weighted degree and adjacency matrices, respectively These gaps, finite analogs to band gaps in continuous systems, enable precise control over the dynamics in a wide range of graph-based physical systems.
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call